# Tagged Questions

Coefficients involved in the Binomial Theorem. $\binom{n}{k}$ counts the subsets of size $k$ of a set of size $n$.

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### prove the formula and then evalute the sum

m,n,r are given non-negative integers, show that $\sum_{k>=-n}$ ${r \choose m+k}$ ${s \choose n+k}$ $=$ ${r+s \choose r-m+n}$ Then evaluate $\sum_{k>=0}k$ ${r \choose k}$ ${s \choose k}$ I ...
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### How to evaluate: $\int_0^1x^{n-1}(1-x)^{n+1}dx$

How can I evaluate the following integral? ($n \in R$, $n>0$) $$\int_0^1x^{n-1}(1-x)^{n+1}dx$$ I was solving the following problem (as practice) in school: Prove that the sum of $n+1$ terms of ...
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### Finding $\binom{999}{0}-\binom{999}{2}+\binom{999}{4}-\binom{999}{6}+\cdots +\binom{999}{996}-\binom{999}{998}$

How to find this alternating sum of binomial coefficients? $$\binom{999}{0}-\binom{999}{2}+\binom{999}{4}-\binom{999}{6}+\cdots +\binom{999}{996}-\binom{999}{998}$$
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### In how many ways can four persons each throwing dice once sum up to 13?

I am solving it by finding out Coefficient of $x^{13}$ in $(x+x^2+....x^6)^4$ but I cannot get the correct answer. Please provide me the final answer if method I am following is correct.
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### relationship between pascal's triangle and number of combinations?

I was able to solve a classic algorithm question, robot paths by using pascal's triangle (PT). This is where a robot starts in the upper left corner and can only go down or right. I kind of reverse ...
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### Determine maximal addend in Newton Binomial Expansion.

Determine the maximal addend in Newton Binomial Expansion of the expression $$\left ( 2n+\frac{1}{2n} \right )^{4n+1},\quad \left ( \forall n \in \mathbb{N} \setminus \left \{ 1 \right \} \right )$$ ...
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### Show that $\frac{n(n-1)(n-2)\times\cdots\times(n-r+1)}{r!}=\frac{n!}{r!(n-r)!}$ = The binomial coefficient formula

I have written in a textbook that $$\cfrac{n(n-1)(n-2)\times\cdots\times(n-r+1)}{r!}\tag{1}$$ $$=\cfrac{n(n-1)(n-2)\times\cdots \times 2 \times 1}{r!(n-r)(n-r-1)\cdots \times 2 \times 1}\tag{2}$$ =\...
### Find the Coefficient of $x^3$ in $(2+x^2)^3 (3+2x)^7$
I'm asked to find the coefficient of $x^3$ in $(2+x^2)^3 (3+2x)^7$. For a simple problem like finding $x^2y^3$ in $(x+y)^5$ I can solve easily using binomial theorem. But I have no idea how to go ...
A friend of mine was doodling with numbers arranged somewhat reminiscent of Pascal's Triangle, where the first row was $1^{n-1} \ 2^{n-1} \dots n^{n-1}$ and subsequent rows were computed by taking ...