Question

Proving $\sum_{i=0}^n (-1)^i\binom{n}{i}\binom{m+i}{m}=(-1)^n\binom{m}{m-n}$

score 101 · Answer 1 · 2026-04-14 23:42:20

101

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Proving $\sum_{i=0}^n (-1)^i\binom{n}{i}\binom{m+i}{m}=(-1)^n\binom{m}{m-n}$

Published on 14 Apr 2026 - 23:42

score 114 · Answer 2 · 2026-04-15 12:07:58

114

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binomial coefficient identity $\sum_{k=0}^n {n\choose k}\sum_{j=0}^{k-1}{{n-1}\choose j}=2^{2n-2}$

Published on 15 Apr 2026 - 12:07

score 64 · Answer 3 · 2026-04-12 10:52:09

64

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Is Wikipedia's inductive proof of the Binomial Theorem acceptable?

Published on 12 Apr 2026 - 10:52

score 85 · Answer 4 · 2026-04-19 04:14:47

85

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Conjecture: $\sum_{m=2}^L \,\,\sum_{l=\max(M+m-L,1)}^{\min(m-1,M)}{L-m \choose M-l}={L \choose M}-1$

Published on 19 Apr 2026 - 4:14

score 24 · Answer 5 · 2026-04-12 18:34:15

24

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Sum of products from $k=1$ to $m$ ( Context: Burton's proof of the binomial theorem)

Published on 12 Apr 2026 - 18:34

score 79 · Answer 6 · 2026-04-11 00:15:41

79

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Evaluating a binomial sum with a probabilistic interpretation

Published on 11 Apr 2026 - 0:15

score 42 · Answer 7 · 2026-04-11 15:37:18

42

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Finding a pole by using a fixed point iteration

Published on 11 Apr 2026 - 15:37

score 64 · Answer 8 · 2026-04-15 21:36:31

64

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high order derivative of product

Published on 15 Apr 2026 - 21:36

score 71 · Answer 9 · 2026-04-13 09:36:01

71

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In how many ways it is possible to take out balls from the basket , such that will take out at least one from each color?

Published on 13 Apr 2026 - 9:36

score 19 · Answer 10 · 2026-04-15 05:48:20

19

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Dropping upper limit element of binomial distribution function index summation

Published on 15 Apr 2026 - 5:48

score 142 · Answer 11 · 2026-04-10 16:19:39

142

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Alternating sum of reciprocals of binomial coefficients

Published on 10 Apr 2026 - 16:19

score 89 · Answer 12 · 2026-04-17 15:05:28

89

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Evaluate using combinatorial argument or otherwise :$\sum_{i=0}^{n-1}\sum_{j=i+1}^{n}\left(j\binom{n}{i}+i\binom{n}{j}\right)$

Published on 17 Apr 2026 - 15:05

score 207 · Answer 13 · 2026-04-15 22:10:29

207

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Showing $ \frac{2^j(2k-j)!}{(k-j)!}=\sum_{l=j}^k\frac{2^ll(2k-l-1)!}{(k-l)!} $

Published on 15 Apr 2026 - 22:10

score 64 · Answer 14 · 2026-03-26 22:55:20

64

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What is the total number of words of length $500$ on $\{a,b\}$ such that the letter $"a"$ appears more than $"b"$ ( without Brute force)?

Published on 26 Mar 2026 - 22:55

score 42 · Answer 15 · 2026-04-15 09:09:07

42

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Show that there is an odd number of binomial coefficients when n is odd

Published on 15 Apr 2026 - 9:09

List Question

Proving $\sum_{i=0}^n (-1)^i\binom{n}{i}\binom{m+i}{m}=(-1)^n\binom{m}{m-n}$

binomial coefficient identity $\sum_{k=0}^n {n\choose k}\sum_{j=0}^{k-1}{{n-1}\choose j}=2^{2n-2}$

Is Wikipedia's inductive proof of the Binomial Theorem acceptable?

Conjecture: $\sum_{m=2}^L \,\,\sum_{l=\max(M+m-L,1)}^{\min(m-1,M)}{L-m \choose M-l}={L \choose M}-1$

Sum of products from $k=1$ to $m$ ( Context: Burton's proof of the binomial theorem)

Evaluating a binomial sum with a probabilistic interpretation

Finding a pole by using a fixed point iteration

high order derivative of product

In how many ways it is possible to take out balls from the basket , such that will take out at least one from each color?

Dropping upper limit element of binomial distribution function index summation

Alternating sum of reciprocals of binomial coefficients

Evaluate using combinatorial argument or otherwise :$\sum_{i=0}^{n-1}\sum_{j=i+1}^{n}\left(j\binom{n}{i}+i\binom{n}{j}\right)$

Showing $ \frac{2^j(2k-j)!}{(k-j)!}=\sum_{l=j}^k\frac{2^ll(2k-l-1)!}{(k-l)!} $

What is the total number of words of length $500$ on $\{a,b\}$ such that the letter $"a"$ appears more than $"b"$ ( without Brute force)?

Show that there is an odd number of binomial coefficients when n is odd

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