Counting Permutations: How many permutations of this set are there?
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Question: Let $n geq 2$ be an even integer. A permutation $a_1; a_2; ldots; a_n$ of the set ${1,2, ldots, n}$ is called awesome if $a_2 = 2a_1$. For example, if $n = 6$, then the permutation $3; 6; 4; 1; 5; 2$ is awesome, whereas the permutation $3; 5; 4; 1; 6; 2$ is not awesome.
How many awesome permutations of the set ${1,2, ldots, n}$ are there?
Answer: $frac{n}{2} cdot (n-2)!$
Attempt:
My understanding was since we need $a_2 = 2a_1$ then $a_1 = a_2/2$. So $a_1$ should be the form $n/2$. For $a_2$, I assumed since $a_1$ was already chosen to be $n$, then $a_2$ should be $n-1$. So the total permutations should be $(n/2) cdot (n-1)!$
combinatorics discrete-mathematics permutations
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add a comment |
$begingroup$
Question: Let $n geq 2$ be an even integer. A permutation $a_1; a_2; ldots; a_n$ of the set ${1,2, ldots, n}$ is called awesome if $a_2 = 2a_1$. For example, if $n = 6$, then the permutation $3; 6; 4; 1; 5; 2$ is awesome, whereas the permutation $3; 5; 4; 1; 6; 2$ is not awesome.
How many awesome permutations of the set ${1,2, ldots, n}$ are there?
Answer: $frac{n}{2} cdot (n-2)!$
Attempt:
My understanding was since we need $a_2 = 2a_1$ then $a_1 = a_2/2$. So $a_1$ should be the form $n/2$. For $a_2$, I assumed since $a_1$ was already chosen to be $n$, then $a_2$ should be $n-1$. So the total permutations should be $(n/2) cdot (n-1)!$
combinatorics discrete-mathematics permutations
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2
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How do you get $a_2 = n-1$ if $a_1 = n$? See the given condition carefully. Also, if $a_1$ is a natural number of the form $frac n2$, then what does this say about $n$? You are also wrongly assigning $n$ to $a_1$ above : it should be to $a_2$.
$endgroup$
– астон вілла олоф мэллбэрг
Nov 29 '18 at 23:26
add a comment |
$begingroup$
Question: Let $n geq 2$ be an even integer. A permutation $a_1; a_2; ldots; a_n$ of the set ${1,2, ldots, n}$ is called awesome if $a_2 = 2a_1$. For example, if $n = 6$, then the permutation $3; 6; 4; 1; 5; 2$ is awesome, whereas the permutation $3; 5; 4; 1; 6; 2$ is not awesome.
How many awesome permutations of the set ${1,2, ldots, n}$ are there?
Answer: $frac{n}{2} cdot (n-2)!$
Attempt:
My understanding was since we need $a_2 = 2a_1$ then $a_1 = a_2/2$. So $a_1$ should be the form $n/2$. For $a_2$, I assumed since $a_1$ was already chosen to be $n$, then $a_2$ should be $n-1$. So the total permutations should be $(n/2) cdot (n-1)!$
combinatorics discrete-mathematics permutations
$endgroup$
Question: Let $n geq 2$ be an even integer. A permutation $a_1; a_2; ldots; a_n$ of the set ${1,2, ldots, n}$ is called awesome if $a_2 = 2a_1$. For example, if $n = 6$, then the permutation $3; 6; 4; 1; 5; 2$ is awesome, whereas the permutation $3; 5; 4; 1; 6; 2$ is not awesome.
How many awesome permutations of the set ${1,2, ldots, n}$ are there?
Answer: $frac{n}{2} cdot (n-2)!$
Attempt:
My understanding was since we need $a_2 = 2a_1$ then $a_1 = a_2/2$. So $a_1$ should be the form $n/2$. For $a_2$, I assumed since $a_1$ was already chosen to be $n$, then $a_2$ should be $n-1$. So the total permutations should be $(n/2) cdot (n-1)!$
combinatorics discrete-mathematics permutations
combinatorics discrete-mathematics permutations
edited Nov 29 '18 at 23:39
N. F. Taussig
44.1k93356
44.1k93356
asked Nov 29 '18 at 23:23
TobyToby
1577
1577
2
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How do you get $a_2 = n-1$ if $a_1 = n$? See the given condition carefully. Also, if $a_1$ is a natural number of the form $frac n2$, then what does this say about $n$? You are also wrongly assigning $n$ to $a_1$ above : it should be to $a_2$.
$endgroup$
– астон вілла олоф мэллбэрг
Nov 29 '18 at 23:26
add a comment |
2
$begingroup$
How do you get $a_2 = n-1$ if $a_1 = n$? See the given condition carefully. Also, if $a_1$ is a natural number of the form $frac n2$, then what does this say about $n$? You are also wrongly assigning $n$ to $a_1$ above : it should be to $a_2$.
$endgroup$
– астон вілла олоф мэллбэрг
Nov 29 '18 at 23:26
2
2
$begingroup$
How do you get $a_2 = n-1$ if $a_1 = n$? See the given condition carefully. Also, if $a_1$ is a natural number of the form $frac n2$, then what does this say about $n$? You are also wrongly assigning $n$ to $a_1$ above : it should be to $a_2$.
$endgroup$
– астон вілла олоф мэллбэрг
Nov 29 '18 at 23:26
$begingroup$
How do you get $a_2 = n-1$ if $a_1 = n$? See the given condition carefully. Also, if $a_1$ is a natural number of the form $frac n2$, then what does this say about $n$? You are also wrongly assigning $n$ to $a_1$ above : it should be to $a_2$.
$endgroup$
– астон вілла олоф мэллбэрг
Nov 29 '18 at 23:26
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
First pick what $a_2$ is. It must be an even number from ${1,2,3,dots,n}$. You should be able to convince yourself that you have exactly $n/2$ options for this step, namely picking from ${2,4,6,8,dots,n}$ since if you were to pick an odd number instead then $a_2/2$ would not be an integer. Now that $a_2$ is selected, you then select $a_1$ and here we have no choices to make. Whatever you chose $a_2$ to be then $a_1$ must be half of that. From there we still have $n-2$ remaining positions to fill with the remaining numbers which can be done in $(n-2)!$ ways.
Let us have a running example of the results of our choices. Suppose that $n=6$ for now and let us display what we know about our permutation and underscores for missing information.
Setup: We have permutation of length six that we know nothing else about:
$$underline{~~~}~~underline{~~~}~~underline{~~~}~~underline{~~~}~~underline{~~~}~~underline{~~~}$$
Step 1 : Pick what $a_2$ is. You have $frac{n}{2}$ choices to make. In our case we can choose $a_2$ to be one of the numbers $2,4,6$. We have $n/2$ options available.
For illustrative purposes suppose that we selected $4$ as our choice. Our permutation currently looks like:
$$underline{~~~}~~underline{~4~}~~underline{~~~}~~underline{~~~}~~underline{~~~}~~underline{~~~}$$
Step 2: Now that we know what $a_2$ looks like, we fill in $a_1$. Whatever $a_2$ happened to be, in order for $a_2=2cdot a_1$ to be true that means that $a_1$ must be half of $a_2$. We have only one option for what $a_1$ looks like since we have already chosen what $a_2$ looks like. Yes, without having knowledge of what $a_2$ is, we would have many choices for $a_1$... however that is not the point. The point is that once $a_2$ has been decided we lose all control over what $a_1$ may be and we are left with only a single option for its value.
In our running example, since we had earlier selected $a_2$ to be $4$, that means that $a_1$ must be half of that, i.e. $2$. Our running example now looks like this:
$$underline{~2~}~~underline{~4~}~~underline{~~~}~~underline{~~~}~~underline{~~~}~~underline{~~~}$$
Step 3: Now, let us choose what $a_3$ is. We cannot repeat whatever was selected for either of $a_2$ or $a_1$, leaving us with $n-2$ choices remaining.
In our running example, $a_3$ may be any of ${1,3,5,6}$ for a total of $n-2=6-2=4$ choices. Let us for illustrative purposes suppose we select $5$ for this value. Our running example now looks like this:
$$underline{~2~}~~underline{~4~}~~underline{~5~}~~underline{~~~}~~underline{~~~}~~underline{~~~}$$
Steps 4 - 6: Continue filling in the next entry in the sequence, making sure not to repeat anything previously selected. These steps have $3,2,1$ options remaining respectively.
Multiplying the number of options available for each step, we get $frac{n}{2}times 1times (n-2)times (n-3)times (n-4)times cdots times 2times 1 = frac{n}{2}(n-2)!$ total arrangements.
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why don't we have no choices for a1 in this case? Shouldn't a1 have n choices? And if it doesn't have any choice, then shouldn't the remaining permutations be (n-1)!? Because only a2 was assigned n/2 choices? I maybe confused about this
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– Toby
Nov 29 '18 at 23:50
1
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@Toby I attempted to extend the explanation a bit more.
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– JMoravitz
Nov 30 '18 at 0:20
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That was a crystal clear explanation, I understand it now! Thank you.
$endgroup$
– Toby
Nov 30 '18 at 0:49
add a comment |
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Not quite. $a_1$ should not necessarily be $frac{n}{2}$; rather, it can be any number which is at most $frac{n}{2}$. For example, $2,4,1,3,6,5$ would be awesome. So there's $frac{n}{2}$ choices for $a_1$ in an awesome permutation, and once this is chosen, only one choice for $a_2$ (because it has to be $2a_1$). The rest of the $n-2$ numbers can be ordered arbitrarily in $(n-2)!$ ways, for a total of $frac{n}{2} (n-2)!$ permutations.
$endgroup$
add a comment |
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2 Answers
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2 Answers
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active
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$begingroup$
First pick what $a_2$ is. It must be an even number from ${1,2,3,dots,n}$. You should be able to convince yourself that you have exactly $n/2$ options for this step, namely picking from ${2,4,6,8,dots,n}$ since if you were to pick an odd number instead then $a_2/2$ would not be an integer. Now that $a_2$ is selected, you then select $a_1$ and here we have no choices to make. Whatever you chose $a_2$ to be then $a_1$ must be half of that. From there we still have $n-2$ remaining positions to fill with the remaining numbers which can be done in $(n-2)!$ ways.
Let us have a running example of the results of our choices. Suppose that $n=6$ for now and let us display what we know about our permutation and underscores for missing information.
Setup: We have permutation of length six that we know nothing else about:
$$underline{~~~}~~underline{~~~}~~underline{~~~}~~underline{~~~}~~underline{~~~}~~underline{~~~}$$
Step 1 : Pick what $a_2$ is. You have $frac{n}{2}$ choices to make. In our case we can choose $a_2$ to be one of the numbers $2,4,6$. We have $n/2$ options available.
For illustrative purposes suppose that we selected $4$ as our choice. Our permutation currently looks like:
$$underline{~~~}~~underline{~4~}~~underline{~~~}~~underline{~~~}~~underline{~~~}~~underline{~~~}$$
Step 2: Now that we know what $a_2$ looks like, we fill in $a_1$. Whatever $a_2$ happened to be, in order for $a_2=2cdot a_1$ to be true that means that $a_1$ must be half of $a_2$. We have only one option for what $a_1$ looks like since we have already chosen what $a_2$ looks like. Yes, without having knowledge of what $a_2$ is, we would have many choices for $a_1$... however that is not the point. The point is that once $a_2$ has been decided we lose all control over what $a_1$ may be and we are left with only a single option for its value.
In our running example, since we had earlier selected $a_2$ to be $4$, that means that $a_1$ must be half of that, i.e. $2$. Our running example now looks like this:
$$underline{~2~}~~underline{~4~}~~underline{~~~}~~underline{~~~}~~underline{~~~}~~underline{~~~}$$
Step 3: Now, let us choose what $a_3$ is. We cannot repeat whatever was selected for either of $a_2$ or $a_1$, leaving us with $n-2$ choices remaining.
In our running example, $a_3$ may be any of ${1,3,5,6}$ for a total of $n-2=6-2=4$ choices. Let us for illustrative purposes suppose we select $5$ for this value. Our running example now looks like this:
$$underline{~2~}~~underline{~4~}~~underline{~5~}~~underline{~~~}~~underline{~~~}~~underline{~~~}$$
Steps 4 - 6: Continue filling in the next entry in the sequence, making sure not to repeat anything previously selected. These steps have $3,2,1$ options remaining respectively.
Multiplying the number of options available for each step, we get $frac{n}{2}times 1times (n-2)times (n-3)times (n-4)times cdots times 2times 1 = frac{n}{2}(n-2)!$ total arrangements.
$endgroup$
$begingroup$
why don't we have no choices for a1 in this case? Shouldn't a1 have n choices? And if it doesn't have any choice, then shouldn't the remaining permutations be (n-1)!? Because only a2 was assigned n/2 choices? I maybe confused about this
$endgroup$
– Toby
Nov 29 '18 at 23:50
1
$begingroup$
@Toby I attempted to extend the explanation a bit more.
$endgroup$
– JMoravitz
Nov 30 '18 at 0:20
$begingroup$
That was a crystal clear explanation, I understand it now! Thank you.
$endgroup$
– Toby
Nov 30 '18 at 0:49
add a comment |
$begingroup$
First pick what $a_2$ is. It must be an even number from ${1,2,3,dots,n}$. You should be able to convince yourself that you have exactly $n/2$ options for this step, namely picking from ${2,4,6,8,dots,n}$ since if you were to pick an odd number instead then $a_2/2$ would not be an integer. Now that $a_2$ is selected, you then select $a_1$ and here we have no choices to make. Whatever you chose $a_2$ to be then $a_1$ must be half of that. From there we still have $n-2$ remaining positions to fill with the remaining numbers which can be done in $(n-2)!$ ways.
Let us have a running example of the results of our choices. Suppose that $n=6$ for now and let us display what we know about our permutation and underscores for missing information.
Setup: We have permutation of length six that we know nothing else about:
$$underline{~~~}~~underline{~~~}~~underline{~~~}~~underline{~~~}~~underline{~~~}~~underline{~~~}$$
Step 1 : Pick what $a_2$ is. You have $frac{n}{2}$ choices to make. In our case we can choose $a_2$ to be one of the numbers $2,4,6$. We have $n/2$ options available.
For illustrative purposes suppose that we selected $4$ as our choice. Our permutation currently looks like:
$$underline{~~~}~~underline{~4~}~~underline{~~~}~~underline{~~~}~~underline{~~~}~~underline{~~~}$$
Step 2: Now that we know what $a_2$ looks like, we fill in $a_1$. Whatever $a_2$ happened to be, in order for $a_2=2cdot a_1$ to be true that means that $a_1$ must be half of $a_2$. We have only one option for what $a_1$ looks like since we have already chosen what $a_2$ looks like. Yes, without having knowledge of what $a_2$ is, we would have many choices for $a_1$... however that is not the point. The point is that once $a_2$ has been decided we lose all control over what $a_1$ may be and we are left with only a single option for its value.
In our running example, since we had earlier selected $a_2$ to be $4$, that means that $a_1$ must be half of that, i.e. $2$. Our running example now looks like this:
$$underline{~2~}~~underline{~4~}~~underline{~~~}~~underline{~~~}~~underline{~~~}~~underline{~~~}$$
Step 3: Now, let us choose what $a_3$ is. We cannot repeat whatever was selected for either of $a_2$ or $a_1$, leaving us with $n-2$ choices remaining.
In our running example, $a_3$ may be any of ${1,3,5,6}$ for a total of $n-2=6-2=4$ choices. Let us for illustrative purposes suppose we select $5$ for this value. Our running example now looks like this:
$$underline{~2~}~~underline{~4~}~~underline{~5~}~~underline{~~~}~~underline{~~~}~~underline{~~~}$$
Steps 4 - 6: Continue filling in the next entry in the sequence, making sure not to repeat anything previously selected. These steps have $3,2,1$ options remaining respectively.
Multiplying the number of options available for each step, we get $frac{n}{2}times 1times (n-2)times (n-3)times (n-4)times cdots times 2times 1 = frac{n}{2}(n-2)!$ total arrangements.
$endgroup$
$begingroup$
why don't we have no choices for a1 in this case? Shouldn't a1 have n choices? And if it doesn't have any choice, then shouldn't the remaining permutations be (n-1)!? Because only a2 was assigned n/2 choices? I maybe confused about this
$endgroup$
– Toby
Nov 29 '18 at 23:50
1
$begingroup$
@Toby I attempted to extend the explanation a bit more.
$endgroup$
– JMoravitz
Nov 30 '18 at 0:20
$begingroup$
That was a crystal clear explanation, I understand it now! Thank you.
$endgroup$
– Toby
Nov 30 '18 at 0:49
add a comment |
$begingroup$
First pick what $a_2$ is. It must be an even number from ${1,2,3,dots,n}$. You should be able to convince yourself that you have exactly $n/2$ options for this step, namely picking from ${2,4,6,8,dots,n}$ since if you were to pick an odd number instead then $a_2/2$ would not be an integer. Now that $a_2$ is selected, you then select $a_1$ and here we have no choices to make. Whatever you chose $a_2$ to be then $a_1$ must be half of that. From there we still have $n-2$ remaining positions to fill with the remaining numbers which can be done in $(n-2)!$ ways.
Let us have a running example of the results of our choices. Suppose that $n=6$ for now and let us display what we know about our permutation and underscores for missing information.
Setup: We have permutation of length six that we know nothing else about:
$$underline{~~~}~~underline{~~~}~~underline{~~~}~~underline{~~~}~~underline{~~~}~~underline{~~~}$$
Step 1 : Pick what $a_2$ is. You have $frac{n}{2}$ choices to make. In our case we can choose $a_2$ to be one of the numbers $2,4,6$. We have $n/2$ options available.
For illustrative purposes suppose that we selected $4$ as our choice. Our permutation currently looks like:
$$underline{~~~}~~underline{~4~}~~underline{~~~}~~underline{~~~}~~underline{~~~}~~underline{~~~}$$
Step 2: Now that we know what $a_2$ looks like, we fill in $a_1$. Whatever $a_2$ happened to be, in order for $a_2=2cdot a_1$ to be true that means that $a_1$ must be half of $a_2$. We have only one option for what $a_1$ looks like since we have already chosen what $a_2$ looks like. Yes, without having knowledge of what $a_2$ is, we would have many choices for $a_1$... however that is not the point. The point is that once $a_2$ has been decided we lose all control over what $a_1$ may be and we are left with only a single option for its value.
In our running example, since we had earlier selected $a_2$ to be $4$, that means that $a_1$ must be half of that, i.e. $2$. Our running example now looks like this:
$$underline{~2~}~~underline{~4~}~~underline{~~~}~~underline{~~~}~~underline{~~~}~~underline{~~~}$$
Step 3: Now, let us choose what $a_3$ is. We cannot repeat whatever was selected for either of $a_2$ or $a_1$, leaving us with $n-2$ choices remaining.
In our running example, $a_3$ may be any of ${1,3,5,6}$ for a total of $n-2=6-2=4$ choices. Let us for illustrative purposes suppose we select $5$ for this value. Our running example now looks like this:
$$underline{~2~}~~underline{~4~}~~underline{~5~}~~underline{~~~}~~underline{~~~}~~underline{~~~}$$
Steps 4 - 6: Continue filling in the next entry in the sequence, making sure not to repeat anything previously selected. These steps have $3,2,1$ options remaining respectively.
Multiplying the number of options available for each step, we get $frac{n}{2}times 1times (n-2)times (n-3)times (n-4)times cdots times 2times 1 = frac{n}{2}(n-2)!$ total arrangements.
$endgroup$
First pick what $a_2$ is. It must be an even number from ${1,2,3,dots,n}$. You should be able to convince yourself that you have exactly $n/2$ options for this step, namely picking from ${2,4,6,8,dots,n}$ since if you were to pick an odd number instead then $a_2/2$ would not be an integer. Now that $a_2$ is selected, you then select $a_1$ and here we have no choices to make. Whatever you chose $a_2$ to be then $a_1$ must be half of that. From there we still have $n-2$ remaining positions to fill with the remaining numbers which can be done in $(n-2)!$ ways.
Let us have a running example of the results of our choices. Suppose that $n=6$ for now and let us display what we know about our permutation and underscores for missing information.
Setup: We have permutation of length six that we know nothing else about:
$$underline{~~~}~~underline{~~~}~~underline{~~~}~~underline{~~~}~~underline{~~~}~~underline{~~~}$$
Step 1 : Pick what $a_2$ is. You have $frac{n}{2}$ choices to make. In our case we can choose $a_2$ to be one of the numbers $2,4,6$. We have $n/2$ options available.
For illustrative purposes suppose that we selected $4$ as our choice. Our permutation currently looks like:
$$underline{~~~}~~underline{~4~}~~underline{~~~}~~underline{~~~}~~underline{~~~}~~underline{~~~}$$
Step 2: Now that we know what $a_2$ looks like, we fill in $a_1$. Whatever $a_2$ happened to be, in order for $a_2=2cdot a_1$ to be true that means that $a_1$ must be half of $a_2$. We have only one option for what $a_1$ looks like since we have already chosen what $a_2$ looks like. Yes, without having knowledge of what $a_2$ is, we would have many choices for $a_1$... however that is not the point. The point is that once $a_2$ has been decided we lose all control over what $a_1$ may be and we are left with only a single option for its value.
In our running example, since we had earlier selected $a_2$ to be $4$, that means that $a_1$ must be half of that, i.e. $2$. Our running example now looks like this:
$$underline{~2~}~~underline{~4~}~~underline{~~~}~~underline{~~~}~~underline{~~~}~~underline{~~~}$$
Step 3: Now, let us choose what $a_3$ is. We cannot repeat whatever was selected for either of $a_2$ or $a_1$, leaving us with $n-2$ choices remaining.
In our running example, $a_3$ may be any of ${1,3,5,6}$ for a total of $n-2=6-2=4$ choices. Let us for illustrative purposes suppose we select $5$ for this value. Our running example now looks like this:
$$underline{~2~}~~underline{~4~}~~underline{~5~}~~underline{~~~}~~underline{~~~}~~underline{~~~}$$
Steps 4 - 6: Continue filling in the next entry in the sequence, making sure not to repeat anything previously selected. These steps have $3,2,1$ options remaining respectively.
Multiplying the number of options available for each step, we get $frac{n}{2}times 1times (n-2)times (n-3)times (n-4)times cdots times 2times 1 = frac{n}{2}(n-2)!$ total arrangements.
edited Nov 30 '18 at 0:19
answered Nov 29 '18 at 23:28
JMoravitzJMoravitz
47.4k33886
47.4k33886
$begingroup$
why don't we have no choices for a1 in this case? Shouldn't a1 have n choices? And if it doesn't have any choice, then shouldn't the remaining permutations be (n-1)!? Because only a2 was assigned n/2 choices? I maybe confused about this
$endgroup$
– Toby
Nov 29 '18 at 23:50
1
$begingroup$
@Toby I attempted to extend the explanation a bit more.
$endgroup$
– JMoravitz
Nov 30 '18 at 0:20
$begingroup$
That was a crystal clear explanation, I understand it now! Thank you.
$endgroup$
– Toby
Nov 30 '18 at 0:49
add a comment |
$begingroup$
why don't we have no choices for a1 in this case? Shouldn't a1 have n choices? And if it doesn't have any choice, then shouldn't the remaining permutations be (n-1)!? Because only a2 was assigned n/2 choices? I maybe confused about this
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– Toby
Nov 29 '18 at 23:50
1
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@Toby I attempted to extend the explanation a bit more.
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– JMoravitz
Nov 30 '18 at 0:20
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That was a crystal clear explanation, I understand it now! Thank you.
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– Toby
Nov 30 '18 at 0:49
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why don't we have no choices for a1 in this case? Shouldn't a1 have n choices? And if it doesn't have any choice, then shouldn't the remaining permutations be (n-1)!? Because only a2 was assigned n/2 choices? I maybe confused about this
$endgroup$
– Toby
Nov 29 '18 at 23:50
$begingroup$
why don't we have no choices for a1 in this case? Shouldn't a1 have n choices? And if it doesn't have any choice, then shouldn't the remaining permutations be (n-1)!? Because only a2 was assigned n/2 choices? I maybe confused about this
$endgroup$
– Toby
Nov 29 '18 at 23:50
1
1
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@Toby I attempted to extend the explanation a bit more.
$endgroup$
– JMoravitz
Nov 30 '18 at 0:20
$begingroup$
@Toby I attempted to extend the explanation a bit more.
$endgroup$
– JMoravitz
Nov 30 '18 at 0:20
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That was a crystal clear explanation, I understand it now! Thank you.
$endgroup$
– Toby
Nov 30 '18 at 0:49
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That was a crystal clear explanation, I understand it now! Thank you.
$endgroup$
– Toby
Nov 30 '18 at 0:49
add a comment |
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Not quite. $a_1$ should not necessarily be $frac{n}{2}$; rather, it can be any number which is at most $frac{n}{2}$. For example, $2,4,1,3,6,5$ would be awesome. So there's $frac{n}{2}$ choices for $a_1$ in an awesome permutation, and once this is chosen, only one choice for $a_2$ (because it has to be $2a_1$). The rest of the $n-2$ numbers can be ordered arbitrarily in $(n-2)!$ ways, for a total of $frac{n}{2} (n-2)!$ permutations.
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add a comment |
$begingroup$
Not quite. $a_1$ should not necessarily be $frac{n}{2}$; rather, it can be any number which is at most $frac{n}{2}$. For example, $2,4,1,3,6,5$ would be awesome. So there's $frac{n}{2}$ choices for $a_1$ in an awesome permutation, and once this is chosen, only one choice for $a_2$ (because it has to be $2a_1$). The rest of the $n-2$ numbers can be ordered arbitrarily in $(n-2)!$ ways, for a total of $frac{n}{2} (n-2)!$ permutations.
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add a comment |
$begingroup$
Not quite. $a_1$ should not necessarily be $frac{n}{2}$; rather, it can be any number which is at most $frac{n}{2}$. For example, $2,4,1,3,6,5$ would be awesome. So there's $frac{n}{2}$ choices for $a_1$ in an awesome permutation, and once this is chosen, only one choice for $a_2$ (because it has to be $2a_1$). The rest of the $n-2$ numbers can be ordered arbitrarily in $(n-2)!$ ways, for a total of $frac{n}{2} (n-2)!$ permutations.
$endgroup$
Not quite. $a_1$ should not necessarily be $frac{n}{2}$; rather, it can be any number which is at most $frac{n}{2}$. For example, $2,4,1,3,6,5$ would be awesome. So there's $frac{n}{2}$ choices for $a_1$ in an awesome permutation, and once this is chosen, only one choice for $a_2$ (because it has to be $2a_1$). The rest of the $n-2$ numbers can be ordered arbitrarily in $(n-2)!$ ways, for a total of $frac{n}{2} (n-2)!$ permutations.
answered Nov 29 '18 at 23:26
plattyplatty
3,370320
3,370320
add a comment |
add a comment |
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How do you get $a_2 = n-1$ if $a_1 = n$? See the given condition carefully. Also, if $a_1$ is a natural number of the form $frac n2$, then what does this say about $n$? You are also wrongly assigning $n$ to $a_1$ above : it should be to $a_2$.
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– астон вілла олоф мэллбэрг
Nov 29 '18 at 23:26