General method to solve Combination problems based on binomial series.

Question

How can I solve these types of questions:

$$\sum_{k=0}^{28} k^2{28 \choose k}$$

$$\sum_{k=0}^{28} 3^k{28 \choose k}$$

Answer 1

Hints: The problems are quite different. For the second problem, a good approach would be to experiment with numbers very much smaller than $28$, such as $1$, $2$, and $3$. There will be a strong pattern, and after seeing the pattern you are likely to find a way to show the pattern continues.

But here is a hint that solves the problem quickly. $$\sum_{0}^{28}\binom{28}{k}3^k=\sum_0^{28} \binom{28}{k}3^k 1^{28-k}.$$ Now think Binomial Theorem.

For the first problem, note that that by the Binomial Theorem, $$(1+x)^{28}=\sum_0^{28} \binom{28}{k}x^k.$$ Take the derivative of both sides. That will not finish things, but it is a good start.

Answer 2

HINT:

For the first, $$k^2\binom nk=\{k(k-1)+k\}\binom nk=k(k-1)\binom nk+k \binom nk$$

$$=k(k-1)\frac{n(n-1)\cdot (n-2)!}{k(k-1)\cdot(k-2)!\{(n-2)-(k-2)\}!}+k \frac{n\cdot (n-1)!}{k\cdot(k-1)!\{(n-1)-(k-1)\}!}$$ $$=n(n-1)\binom{n-2}{k-2}+n\binom{n-1}{k-1}$$ if integer $k>1$

Put $k=2,\cdots,n-1,n$

In general, $$(a_0+a_1k+a_2k^2+a_3k^3+\cdots+a_mk^m)\binom nk$$

$$=b_0 \binom nk+ b_1\cdot n\binom{n-1}{k-1}+b_2 n(n-1)\binom{n-2}{k-2}+\cdots +b_m n(n-1)\cdots(n-m+1)\binom nm$$

Multiply either sides by $k!$ and compare the coefficients of the different powers of $k$ to find $b_i$s in terms of $a_i$