New Year Maths $2019$
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$$;;;color{red}{binom {20}{19}}\
color{orange}{+binom {19}{a}
+{binom ab}}\
color{green}{+binom bc
+binom cd}\
color{blue}{+binom de+
binom ef}\
color{purple}{+binom fg+binom gh}\
color{magenta}{+binom hk}\[15pt]
color{orange}=\[10pt]
color{red}{2019}\[15pt]
color{orange}{a,b,}
color{green}{c,d,}
color{blue}{e,f,}
color{purple}{g,h,}
color{magenta}kcolor{orange}{=?}\[20pt]$$
$$color{red}{text{Happy New Year!}}$$
Note - Please feel free to post any other interesting identities based on the same theme!
combinatorics summation binomial-coefficients recreational-mathematics
$endgroup$
|
show 9 more comments
$begingroup$
$$;;;color{red}{binom {20}{19}}\
color{orange}{+binom {19}{a}
+{binom ab}}\
color{green}{+binom bc
+binom cd}\
color{blue}{+binom de+
binom ef}\
color{purple}{+binom fg+binom gh}\
color{magenta}{+binom hk}\[15pt]
color{orange}=\[10pt]
color{red}{2019}\[15pt]
color{orange}{a,b,}
color{green}{c,d,}
color{blue}{e,f,}
color{purple}{g,h,}
color{magenta}kcolor{orange}{=?}\[20pt]$$
$$color{red}{text{Happy New Year!}}$$
Note - Please feel free to post any other interesting identities based on the same theme!
combinatorics summation binomial-coefficients recreational-mathematics
$endgroup$
1
$begingroup$
i think you're missing the g,h term. Else it is a bit too easy
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– Anvit
Dec 30 '18 at 16:06
1
$begingroup$
I was missing you ! Happy New Year !
$endgroup$
– Claude Leibovici
Dec 30 '18 at 16:12
2
$begingroup$
Is python allowed, or is that cheating? :P
$endgroup$
– Anvit
Dec 30 '18 at 16:28
2
$begingroup$
@Anvit - Good question! $i, j$ are usually used for indexing, and probably for the same reason, elements in the standard $3times 3$ matrix are often labelled $left[begin{array}&a&b&c\d&e&f\g&h&kend{array}right]$.
$endgroup$
– hypergeometric
Dec 30 '18 at 17:08
2
$begingroup$
There are $19$ up-votes.... Should I?
$endgroup$
– clathratus
Feb 14 at 1:00
|
show 9 more comments
$begingroup$
$$;;;color{red}{binom {20}{19}}\
color{orange}{+binom {19}{a}
+{binom ab}}\
color{green}{+binom bc
+binom cd}\
color{blue}{+binom de+
binom ef}\
color{purple}{+binom fg+binom gh}\
color{magenta}{+binom hk}\[15pt]
color{orange}=\[10pt]
color{red}{2019}\[15pt]
color{orange}{a,b,}
color{green}{c,d,}
color{blue}{e,f,}
color{purple}{g,h,}
color{magenta}kcolor{orange}{=?}\[20pt]$$
$$color{red}{text{Happy New Year!}}$$
Note - Please feel free to post any other interesting identities based on the same theme!
combinatorics summation binomial-coefficients recreational-mathematics
$endgroup$
$$;;;color{red}{binom {20}{19}}\
color{orange}{+binom {19}{a}
+{binom ab}}\
color{green}{+binom bc
+binom cd}\
color{blue}{+binom de+
binom ef}\
color{purple}{+binom fg+binom gh}\
color{magenta}{+binom hk}\[15pt]
color{orange}=\[10pt]
color{red}{2019}\[15pt]
color{orange}{a,b,}
color{green}{c,d,}
color{blue}{e,f,}
color{purple}{g,h,}
color{magenta}kcolor{orange}{=?}\[20pt]$$
$$color{red}{text{Happy New Year!}}$$
Note - Please feel free to post any other interesting identities based on the same theme!
combinatorics summation binomial-coefficients recreational-mathematics
combinatorics summation binomial-coefficients recreational-mathematics
edited Dec 31 '18 at 9:51
community wiki
hypergeometric
1
$begingroup$
i think you're missing the g,h term. Else it is a bit too easy
$endgroup$
– Anvit
Dec 30 '18 at 16:06
1
$begingroup$
I was missing you ! Happy New Year !
$endgroup$
– Claude Leibovici
Dec 30 '18 at 16:12
2
$begingroup$
Is python allowed, or is that cheating? :P
$endgroup$
– Anvit
Dec 30 '18 at 16:28
2
$begingroup$
@Anvit - Good question! $i, j$ are usually used for indexing, and probably for the same reason, elements in the standard $3times 3$ matrix are often labelled $left[begin{array}&a&b&c\d&e&f\g&h&kend{array}right]$.
$endgroup$
– hypergeometric
Dec 30 '18 at 17:08
2
$begingroup$
There are $19$ up-votes.... Should I?
$endgroup$
– clathratus
Feb 14 at 1:00
|
show 9 more comments
1
$begingroup$
i think you're missing the g,h term. Else it is a bit too easy
$endgroup$
– Anvit
Dec 30 '18 at 16:06
1
$begingroup$
I was missing you ! Happy New Year !
$endgroup$
– Claude Leibovici
Dec 30 '18 at 16:12
2
$begingroup$
Is python allowed, or is that cheating? :P
$endgroup$
– Anvit
Dec 30 '18 at 16:28
2
$begingroup$
@Anvit - Good question! $i, j$ are usually used for indexing, and probably for the same reason, elements in the standard $3times 3$ matrix are often labelled $left[begin{array}&a&b&c\d&e&f\g&h&kend{array}right]$.
$endgroup$
– hypergeometric
Dec 30 '18 at 17:08
2
$begingroup$
There are $19$ up-votes.... Should I?
$endgroup$
– clathratus
Feb 14 at 1:00
1
1
$begingroup$
i think you're missing the g,h term. Else it is a bit too easy
$endgroup$
– Anvit
Dec 30 '18 at 16:06
$begingroup$
i think you're missing the g,h term. Else it is a bit too easy
$endgroup$
– Anvit
Dec 30 '18 at 16:06
1
1
$begingroup$
I was missing you ! Happy New Year !
$endgroup$
– Claude Leibovici
Dec 30 '18 at 16:12
$begingroup$
I was missing you ! Happy New Year !
$endgroup$
– Claude Leibovici
Dec 30 '18 at 16:12
2
2
$begingroup$
Is python allowed, or is that cheating? :P
$endgroup$
– Anvit
Dec 30 '18 at 16:28
$begingroup$
Is python allowed, or is that cheating? :P
$endgroup$
– Anvit
Dec 30 '18 at 16:28
2
2
$begingroup$
@Anvit - Good question! $i, j$ are usually used for indexing, and probably for the same reason, elements in the standard $3times 3$ matrix are often labelled $left[begin{array}&a&b&c\d&e&f\g&h&kend{array}right]$.
$endgroup$
– hypergeometric
Dec 30 '18 at 17:08
$begingroup$
@Anvit - Good question! $i, j$ are usually used for indexing, and probably for the same reason, elements in the standard $3times 3$ matrix are often labelled $left[begin{array}&a&b&c\d&e&f\g&h&kend{array}right]$.
$endgroup$
– hypergeometric
Dec 30 '18 at 17:08
2
2
$begingroup$
There are $19$ up-votes.... Should I?
$endgroup$
– clathratus
Feb 14 at 1:00
$begingroup$
There are $19$ up-votes.... Should I?
$endgroup$
– clathratus
Feb 14 at 1:00
|
show 9 more comments
5 Answers
5
active
oldest
votes
$begingroup$
A simple python script yielded the following results for a,b,c,d,e,f,g,h,k such that $19>a> b> c> cdots > kgeq 0$
16,14,12,7,6,5,3,1,0
16,14,12,7,6,5,3,2,0
17,14,10,7,6,5,3,1,0
17,14,10,7,6,5,3,2,0
18,15,14,10,9,5,4,2,0
18,15,14,10,9,5,4,3,1
18,15,14,10,9,5,4,3,2
18,17,16,12,10,9,7,6,1
18,17,16,12,10,9,7,6,5
18,17,16,12,11,9,7,2,0
18,17,16,12,11,9,7,5,0
18,17,16,12,11,9,7,6,2
18,17,16,12,11,9,7,6,4
18,17,16,12,11,10,8,3,0
18,17,16,12,11,10,8,5,0
$endgroup$
$begingroup$
Nice and complete solution! (+1)
$endgroup$
– hypergeometric
Dec 31 '18 at 9:22
add a comment |
$begingroup$
$$color{red}{binom {20}{19}}
+color{orange}{binom {19}{16}
+{binom {16}{14}}}+
color{green}{binom {14}{12}
+binom {12}{7}}+
color{blue}{binom {7}{6}+
binom {6}{5}}+
color{purple}{binom {5}{3}+binom {3}{1}}+
color{magenta}{binom {1}{0}}
=color{red}{2019}$$
$endgroup$
3
$begingroup$
Nice! We can make each of JMoravitz's solutions into a tree and and then vote which tree is the most beautiful ...
$endgroup$
– Bram28
Dec 30 '18 at 22:46
3
$begingroup$
Very nice animation! (+1)
$endgroup$
– hypergeometric
Dec 31 '18 at 9:21
add a comment |
$begingroup$
$$color{red}{binom{20}{19}}+color{orange}{binom{19}{18}}+color{orange}{binom{18}{17}}+color{green}{binom{17}{16}}+color{green}{binom{16}{12}}+color{blue}{binom{12}{11}}+color{blue}{binom{11}{10}}+color{purple}{binom{10}{8}}+color{purple}{binom{8}{5}}+color{magenta}{binom{5}{0}}color{orange}=color{red}{2019}$$
$color{red}{text{Happy New Year!}}$
$endgroup$
$begingroup$
Nice! Hope you don't mind some colors :D
$endgroup$
– Zacky
Dec 31 '18 at 11:59
2
$begingroup$
Thank you, Zacky!
$endgroup$
– Michael Rozenberg
Dec 31 '18 at 12:27
add a comment |
$begingroup$
$$begin{align}&{7choose H}+color{red}{9choose A}+color{blue}{11choose P}+{color{blue}{12choose P}}+color{green}{13choose Y}+\
&{8choose N}+color{magenta}{16choose E}+{15choose W}+\
&color{green}{14choose Y}+color{magenta}{17choose E}+color{red}{10choose A}+binom{18}{R}=
2019.end{align}$$
$binom76+binom85+binom98+binom{10}8+binom{11}{10}+binom{12}{10}+binom{13}{12}+binom{14}{12}+binom{15}{13}+binom{16}{14}+binom{17}{14}+binom{18}{15}=2019.$
$endgroup$
$begingroup$
Cool! ... too bad that the same letters don't go with the same numbers on top ... did you try that?
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– Bram28
Dec 30 '18 at 22:39
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Nope, first I tried $binom7{H}+binom8{A}+cdots+binom{18}{R}$, but then had to adjust (by trial and error, no coding).
$endgroup$
– farruhota
Dec 30 '18 at 22:54
add a comment |
$begingroup$
$color{red}{{20 choose 19}+{19choose 16}}+color{orange}{{16choose 3}+{3choose 1}}+color{green}{{1choose 2}+{2choose 2}}+color{blue}{{2choose 466}+{466choose 465}}+color{purple}{{465choose 466}+{466choose 467}}=color{magenta}{2019}$.
$endgroup$
add a comment |
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5 Answers
5
active
oldest
votes
5 Answers
5
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
A simple python script yielded the following results for a,b,c,d,e,f,g,h,k such that $19>a> b> c> cdots > kgeq 0$
16,14,12,7,6,5,3,1,0
16,14,12,7,6,5,3,2,0
17,14,10,7,6,5,3,1,0
17,14,10,7,6,5,3,2,0
18,15,14,10,9,5,4,2,0
18,15,14,10,9,5,4,3,1
18,15,14,10,9,5,4,3,2
18,17,16,12,10,9,7,6,1
18,17,16,12,10,9,7,6,5
18,17,16,12,11,9,7,2,0
18,17,16,12,11,9,7,5,0
18,17,16,12,11,9,7,6,2
18,17,16,12,11,9,7,6,4
18,17,16,12,11,10,8,3,0
18,17,16,12,11,10,8,5,0
$endgroup$
$begingroup$
Nice and complete solution! (+1)
$endgroup$
– hypergeometric
Dec 31 '18 at 9:22
add a comment |
$begingroup$
A simple python script yielded the following results for a,b,c,d,e,f,g,h,k such that $19>a> b> c> cdots > kgeq 0$
16,14,12,7,6,5,3,1,0
16,14,12,7,6,5,3,2,0
17,14,10,7,6,5,3,1,0
17,14,10,7,6,5,3,2,0
18,15,14,10,9,5,4,2,0
18,15,14,10,9,5,4,3,1
18,15,14,10,9,5,4,3,2
18,17,16,12,10,9,7,6,1
18,17,16,12,10,9,7,6,5
18,17,16,12,11,9,7,2,0
18,17,16,12,11,9,7,5,0
18,17,16,12,11,9,7,6,2
18,17,16,12,11,9,7,6,4
18,17,16,12,11,10,8,3,0
18,17,16,12,11,10,8,5,0
$endgroup$
$begingroup$
Nice and complete solution! (+1)
$endgroup$
– hypergeometric
Dec 31 '18 at 9:22
add a comment |
$begingroup$
A simple python script yielded the following results for a,b,c,d,e,f,g,h,k such that $19>a> b> c> cdots > kgeq 0$
16,14,12,7,6,5,3,1,0
16,14,12,7,6,5,3,2,0
17,14,10,7,6,5,3,1,0
17,14,10,7,6,5,3,2,0
18,15,14,10,9,5,4,2,0
18,15,14,10,9,5,4,3,1
18,15,14,10,9,5,4,3,2
18,17,16,12,10,9,7,6,1
18,17,16,12,10,9,7,6,5
18,17,16,12,11,9,7,2,0
18,17,16,12,11,9,7,5,0
18,17,16,12,11,9,7,6,2
18,17,16,12,11,9,7,6,4
18,17,16,12,11,10,8,3,0
18,17,16,12,11,10,8,5,0
$endgroup$
A simple python script yielded the following results for a,b,c,d,e,f,g,h,k such that $19>a> b> c> cdots > kgeq 0$
16,14,12,7,6,5,3,1,0
16,14,12,7,6,5,3,2,0
17,14,10,7,6,5,3,1,0
17,14,10,7,6,5,3,2,0
18,15,14,10,9,5,4,2,0
18,15,14,10,9,5,4,3,1
18,15,14,10,9,5,4,3,2
18,17,16,12,10,9,7,6,1
18,17,16,12,10,9,7,6,5
18,17,16,12,11,9,7,2,0
18,17,16,12,11,9,7,5,0
18,17,16,12,11,9,7,6,2
18,17,16,12,11,9,7,6,4
18,17,16,12,11,10,8,3,0
18,17,16,12,11,10,8,5,0
edited Dec 30 '18 at 17:39
community wiki
JMoravitz
$begingroup$
Nice and complete solution! (+1)
$endgroup$
– hypergeometric
Dec 31 '18 at 9:22
add a comment |
$begingroup$
Nice and complete solution! (+1)
$endgroup$
– hypergeometric
Dec 31 '18 at 9:22
$begingroup$
Nice and complete solution! (+1)
$endgroup$
– hypergeometric
Dec 31 '18 at 9:22
$begingroup$
Nice and complete solution! (+1)
$endgroup$
– hypergeometric
Dec 31 '18 at 9:22
add a comment |
$begingroup$
$$color{red}{binom {20}{19}}
+color{orange}{binom {19}{16}
+{binom {16}{14}}}+
color{green}{binom {14}{12}
+binom {12}{7}}+
color{blue}{binom {7}{6}+
binom {6}{5}}+
color{purple}{binom {5}{3}+binom {3}{1}}+
color{magenta}{binom {1}{0}}
=color{red}{2019}$$
$endgroup$
3
$begingroup$
Nice! We can make each of JMoravitz's solutions into a tree and and then vote which tree is the most beautiful ...
$endgroup$
– Bram28
Dec 30 '18 at 22:46
3
$begingroup$
Very nice animation! (+1)
$endgroup$
– hypergeometric
Dec 31 '18 at 9:21
add a comment |
$begingroup$
$$color{red}{binom {20}{19}}
+color{orange}{binom {19}{16}
+{binom {16}{14}}}+
color{green}{binom {14}{12}
+binom {12}{7}}+
color{blue}{binom {7}{6}+
binom {6}{5}}+
color{purple}{binom {5}{3}+binom {3}{1}}+
color{magenta}{binom {1}{0}}
=color{red}{2019}$$
$endgroup$
3
$begingroup$
Nice! We can make each of JMoravitz's solutions into a tree and and then vote which tree is the most beautiful ...
$endgroup$
– Bram28
Dec 30 '18 at 22:46
3
$begingroup$
Very nice animation! (+1)
$endgroup$
– hypergeometric
Dec 31 '18 at 9:21
add a comment |
$begingroup$
$$color{red}{binom {20}{19}}
+color{orange}{binom {19}{16}
+{binom {16}{14}}}+
color{green}{binom {14}{12}
+binom {12}{7}}+
color{blue}{binom {7}{6}+
binom {6}{5}}+
color{purple}{binom {5}{3}+binom {3}{1}}+
color{magenta}{binom {1}{0}}
=color{red}{2019}$$
$endgroup$
$$color{red}{binom {20}{19}}
+color{orange}{binom {19}{16}
+{binom {16}{14}}}+
color{green}{binom {14}{12}
+binom {12}{7}}+
color{blue}{binom {7}{6}+
binom {6}{5}}+
color{purple}{binom {5}{3}+binom {3}{1}}+
color{magenta}{binom {1}{0}}
=color{red}{2019}$$
edited Dec 30 '18 at 22:34
community wiki
caverac
3
$begingroup$
Nice! We can make each of JMoravitz's solutions into a tree and and then vote which tree is the most beautiful ...
$endgroup$
– Bram28
Dec 30 '18 at 22:46
3
$begingroup$
Very nice animation! (+1)
$endgroup$
– hypergeometric
Dec 31 '18 at 9:21
add a comment |
3
$begingroup$
Nice! We can make each of JMoravitz's solutions into a tree and and then vote which tree is the most beautiful ...
$endgroup$
– Bram28
Dec 30 '18 at 22:46
3
$begingroup$
Very nice animation! (+1)
$endgroup$
– hypergeometric
Dec 31 '18 at 9:21
3
3
$begingroup$
Nice! We can make each of JMoravitz's solutions into a tree and and then vote which tree is the most beautiful ...
$endgroup$
– Bram28
Dec 30 '18 at 22:46
$begingroup$
Nice! We can make each of JMoravitz's solutions into a tree and and then vote which tree is the most beautiful ...
$endgroup$
– Bram28
Dec 30 '18 at 22:46
3
3
$begingroup$
Very nice animation! (+1)
$endgroup$
– hypergeometric
Dec 31 '18 at 9:21
$begingroup$
Very nice animation! (+1)
$endgroup$
– hypergeometric
Dec 31 '18 at 9:21
add a comment |
$begingroup$
$$color{red}{binom{20}{19}}+color{orange}{binom{19}{18}}+color{orange}{binom{18}{17}}+color{green}{binom{17}{16}}+color{green}{binom{16}{12}}+color{blue}{binom{12}{11}}+color{blue}{binom{11}{10}}+color{purple}{binom{10}{8}}+color{purple}{binom{8}{5}}+color{magenta}{binom{5}{0}}color{orange}=color{red}{2019}$$
$color{red}{text{Happy New Year!}}$
$endgroup$
$begingroup$
Nice! Hope you don't mind some colors :D
$endgroup$
– Zacky
Dec 31 '18 at 11:59
2
$begingroup$
Thank you, Zacky!
$endgroup$
– Michael Rozenberg
Dec 31 '18 at 12:27
add a comment |
$begingroup$
$$color{red}{binom{20}{19}}+color{orange}{binom{19}{18}}+color{orange}{binom{18}{17}}+color{green}{binom{17}{16}}+color{green}{binom{16}{12}}+color{blue}{binom{12}{11}}+color{blue}{binom{11}{10}}+color{purple}{binom{10}{8}}+color{purple}{binom{8}{5}}+color{magenta}{binom{5}{0}}color{orange}=color{red}{2019}$$
$color{red}{text{Happy New Year!}}$
$endgroup$
$begingroup$
Nice! Hope you don't mind some colors :D
$endgroup$
– Zacky
Dec 31 '18 at 11:59
2
$begingroup$
Thank you, Zacky!
$endgroup$
– Michael Rozenberg
Dec 31 '18 at 12:27
add a comment |
$begingroup$
$$color{red}{binom{20}{19}}+color{orange}{binom{19}{18}}+color{orange}{binom{18}{17}}+color{green}{binom{17}{16}}+color{green}{binom{16}{12}}+color{blue}{binom{12}{11}}+color{blue}{binom{11}{10}}+color{purple}{binom{10}{8}}+color{purple}{binom{8}{5}}+color{magenta}{binom{5}{0}}color{orange}=color{red}{2019}$$
$color{red}{text{Happy New Year!}}$
$endgroup$
$$color{red}{binom{20}{19}}+color{orange}{binom{19}{18}}+color{orange}{binom{18}{17}}+color{green}{binom{17}{16}}+color{green}{binom{16}{12}}+color{blue}{binom{12}{11}}+color{blue}{binom{11}{10}}+color{purple}{binom{10}{8}}+color{purple}{binom{8}{5}}+color{magenta}{binom{5}{0}}color{orange}=color{red}{2019}$$
$color{red}{text{Happy New Year!}}$
edited Dec 31 '18 at 11:58
community wiki
Michael Rozenberg
$begingroup$
Nice! Hope you don't mind some colors :D
$endgroup$
– Zacky
Dec 31 '18 at 11:59
2
$begingroup$
Thank you, Zacky!
$endgroup$
– Michael Rozenberg
Dec 31 '18 at 12:27
add a comment |
$begingroup$
Nice! Hope you don't mind some colors :D
$endgroup$
– Zacky
Dec 31 '18 at 11:59
2
$begingroup$
Thank you, Zacky!
$endgroup$
– Michael Rozenberg
Dec 31 '18 at 12:27
$begingroup$
Nice! Hope you don't mind some colors :D
$endgroup$
– Zacky
Dec 31 '18 at 11:59
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Nice! Hope you don't mind some colors :D
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– Zacky
Dec 31 '18 at 11:59
2
2
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Thank you, Zacky!
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– Michael Rozenberg
Dec 31 '18 at 12:27
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Thank you, Zacky!
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– Michael Rozenberg
Dec 31 '18 at 12:27
add a comment |
$begingroup$
$$begin{align}&{7choose H}+color{red}{9choose A}+color{blue}{11choose P}+{color{blue}{12choose P}}+color{green}{13choose Y}+\
&{8choose N}+color{magenta}{16choose E}+{15choose W}+\
&color{green}{14choose Y}+color{magenta}{17choose E}+color{red}{10choose A}+binom{18}{R}=
2019.end{align}$$
$binom76+binom85+binom98+binom{10}8+binom{11}{10}+binom{12}{10}+binom{13}{12}+binom{14}{12}+binom{15}{13}+binom{16}{14}+binom{17}{14}+binom{18}{15}=2019.$
$endgroup$
$begingroup$
Cool! ... too bad that the same letters don't go with the same numbers on top ... did you try that?
$endgroup$
– Bram28
Dec 30 '18 at 22:39
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Nope, first I tried $binom7{H}+binom8{A}+cdots+binom{18}{R}$, but then had to adjust (by trial and error, no coding).
$endgroup$
– farruhota
Dec 30 '18 at 22:54
add a comment |
$begingroup$
$$begin{align}&{7choose H}+color{red}{9choose A}+color{blue}{11choose P}+{color{blue}{12choose P}}+color{green}{13choose Y}+\
&{8choose N}+color{magenta}{16choose E}+{15choose W}+\
&color{green}{14choose Y}+color{magenta}{17choose E}+color{red}{10choose A}+binom{18}{R}=
2019.end{align}$$
$binom76+binom85+binom98+binom{10}8+binom{11}{10}+binom{12}{10}+binom{13}{12}+binom{14}{12}+binom{15}{13}+binom{16}{14}+binom{17}{14}+binom{18}{15}=2019.$
$endgroup$
$begingroup$
Cool! ... too bad that the same letters don't go with the same numbers on top ... did you try that?
$endgroup$
– Bram28
Dec 30 '18 at 22:39
$begingroup$
Nope, first I tried $binom7{H}+binom8{A}+cdots+binom{18}{R}$, but then had to adjust (by trial and error, no coding).
$endgroup$
– farruhota
Dec 30 '18 at 22:54
add a comment |
$begingroup$
$$begin{align}&{7choose H}+color{red}{9choose A}+color{blue}{11choose P}+{color{blue}{12choose P}}+color{green}{13choose Y}+\
&{8choose N}+color{magenta}{16choose E}+{15choose W}+\
&color{green}{14choose Y}+color{magenta}{17choose E}+color{red}{10choose A}+binom{18}{R}=
2019.end{align}$$
$binom76+binom85+binom98+binom{10}8+binom{11}{10}+binom{12}{10}+binom{13}{12}+binom{14}{12}+binom{15}{13}+binom{16}{14}+binom{17}{14}+binom{18}{15}=2019.$
$endgroup$
$$begin{align}&{7choose H}+color{red}{9choose A}+color{blue}{11choose P}+{color{blue}{12choose P}}+color{green}{13choose Y}+\
&{8choose N}+color{magenta}{16choose E}+{15choose W}+\
&color{green}{14choose Y}+color{magenta}{17choose E}+color{red}{10choose A}+binom{18}{R}=
2019.end{align}$$
$binom76+binom85+binom98+binom{10}8+binom{11}{10}+binom{12}{10}+binom{13}{12}+binom{14}{12}+binom{15}{13}+binom{16}{14}+binom{17}{14}+binom{18}{15}=2019.$
answered Dec 30 '18 at 22:33
community wiki
farruhota
$begingroup$
Cool! ... too bad that the same letters don't go with the same numbers on top ... did you try that?
$endgroup$
– Bram28
Dec 30 '18 at 22:39
$begingroup$
Nope, first I tried $binom7{H}+binom8{A}+cdots+binom{18}{R}$, but then had to adjust (by trial and error, no coding).
$endgroup$
– farruhota
Dec 30 '18 at 22:54
add a comment |
$begingroup$
Cool! ... too bad that the same letters don't go with the same numbers on top ... did you try that?
$endgroup$
– Bram28
Dec 30 '18 at 22:39
$begingroup$
Nope, first I tried $binom7{H}+binom8{A}+cdots+binom{18}{R}$, but then had to adjust (by trial and error, no coding).
$endgroup$
– farruhota
Dec 30 '18 at 22:54
$begingroup$
Cool! ... too bad that the same letters don't go with the same numbers on top ... did you try that?
$endgroup$
– Bram28
Dec 30 '18 at 22:39
$begingroup$
Cool! ... too bad that the same letters don't go with the same numbers on top ... did you try that?
$endgroup$
– Bram28
Dec 30 '18 at 22:39
$begingroup$
Nope, first I tried $binom7{H}+binom8{A}+cdots+binom{18}{R}$, but then had to adjust (by trial and error, no coding).
$endgroup$
– farruhota
Dec 30 '18 at 22:54
$begingroup$
Nope, first I tried $binom7{H}+binom8{A}+cdots+binom{18}{R}$, but then had to adjust (by trial and error, no coding).
$endgroup$
– farruhota
Dec 30 '18 at 22:54
add a comment |
$begingroup$
$color{red}{{20 choose 19}+{19choose 16}}+color{orange}{{16choose 3}+{3choose 1}}+color{green}{{1choose 2}+{2choose 2}}+color{blue}{{2choose 466}+{466choose 465}}+color{purple}{{465choose 466}+{466choose 467}}=color{magenta}{2019}$.
$endgroup$
add a comment |
$begingroup$
$color{red}{{20 choose 19}+{19choose 16}}+color{orange}{{16choose 3}+{3choose 1}}+color{green}{{1choose 2}+{2choose 2}}+color{blue}{{2choose 466}+{466choose 465}}+color{purple}{{465choose 466}+{466choose 467}}=color{magenta}{2019}$.
$endgroup$
add a comment |
$begingroup$
$color{red}{{20 choose 19}+{19choose 16}}+color{orange}{{16choose 3}+{3choose 1}}+color{green}{{1choose 2}+{2choose 2}}+color{blue}{{2choose 466}+{466choose 465}}+color{purple}{{465choose 466}+{466choose 467}}=color{magenta}{2019}$.
$endgroup$
$color{red}{{20 choose 19}+{19choose 16}}+color{orange}{{16choose 3}+{3choose 1}}+color{green}{{1choose 2}+{2choose 2}}+color{blue}{{2choose 466}+{466choose 465}}+color{purple}{{465choose 466}+{466choose 467}}=color{magenta}{2019}$.
edited Jan 1 at 6:11
community wiki
Chris Custer
add a comment |
add a comment |
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1
$begingroup$
i think you're missing the g,h term. Else it is a bit too easy
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– Anvit
Dec 30 '18 at 16:06
1
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I was missing you ! Happy New Year !
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– Claude Leibovici
Dec 30 '18 at 16:12
2
$begingroup$
Is python allowed, or is that cheating? :P
$endgroup$
– Anvit
Dec 30 '18 at 16:28
2
$begingroup$
@Anvit - Good question! $i, j$ are usually used for indexing, and probably for the same reason, elements in the standard $3times 3$ matrix are often labelled $left[begin{array}&a&b&c\d&e&f\g&h&kend{array}right]$.
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– hypergeometric
Dec 30 '18 at 17:08
2
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There are $19$ up-votes.... Should I?
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– clathratus
Feb 14 at 1:00