Proving Identity In Combinatorial and Algebraic Way

Question

I need to prove the identity $$\sum_{k = 0}^{n} k^2\binom{n}{k} = n(n - 1)2^{n - 2} + n2^{n - 1}$$ in two ways : in algebraic way and combinatorial way.

About the algebraic way - I tried to find a formula to be placed in the identity that could help me solve the problem but without success. Combinatorial way - I couldn't find a "story" that is proof of identity.

Take a committee of $n$ people. A subcommittee is formed; it must have a chair and secretary, but they can be the same person. — Angina Seng, Apr 07 '19 at 17:28
In your summation, $n \choose k$ is the number of ways to pick a subset of size $k$ from a set of size $n$. You now have $k$ distinguished elements. Think about what the $k^2$ means in terms of these distinguished elements. — Jeremy Dover, Apr 07 '19 at 17:28
Welcome to MathSE. This tutorial explains how to typeset mathematics on this site. — N. F. Taussig, Apr 07 '19 at 17:31
For the algebraic route, one attack is to try an manipulate the binomial theorem $(1+x)^n=\dots$, taking derivatives and such to get the sum you want. Another is to write $k^2=k(k-1)+k$, split into two sums, and use the absorption identity. — Mike Earnest, Apr 07 '19 at 17:49
Possible duplicate of Induction: $\sum_{k=0}^n \binom nk k^2 = n(1+n)2^{n-2}$ (which has algebraic as well as combinatorial answers) — Sil, Apr 07 '19 at 18:42
@ISR_Student: Check out Lord Shark's comment. There are $n \choose k$ ways to form the subcommittee. The $k^2$ corresponds to the chair and secretary. — Jeremy Dover, Apr 07 '19 at 20:30

score 0 · Answer 1 · answered Apr 07 '19 at 17:59

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Algebraic way: Consider the function $f(x)=(x+1)^n=\sum_{k=0}^{n}{{n\choose k}}x^k$. Note that $f(1)=2^n$.

Now we differentiate twice: $$f’(1)=n(1+1)^{n-1}= \sum_{k=1}^{n}{k{n\choose k}} $$ and $$f’’(1)=n(n-1)(1+1)^{n-2}= \sum_{k=2}^{n}{(k^2-k){n\choose k}}$$ $$= f’’(1)=n(n-1)2^{n-2}= \sum_{k=2}^{n}{k^2{n\choose k}}-\sum_{k=2}^{n}{k{n\choose k}}$$ $$n(n-1)2^{n-2}= \sum_{k=2}^{n}{k^2{n\choose k}}-f’(1)+n$$ Can you take it from here?

answered Apr 07 '19 at 17:59

DINEDINE

6,081

'n' how to i get k from 0 to k in the sigma in the right side , it connects with 'n' that also in the right side of the final equation? – ISR_Student Apr 07 '19 at 18:07
I don’t understand what you mean – DINEDINE Apr 07 '19 at 18:35

score 0 · Answer 2 · answered Apr 07 '19 at 18:02

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Hint: A combinatorial approach is mentioned in a comment. Here is a hint, try to use:

$$\binom{n}{k}=\frac{n}{k}\binom{n-1}{k-1}$$

answered Apr 07 '19 at 18:02

tarit goswami

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Proving Identity In Combinatorial and Algebraic Way

2 Answers2