# 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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### calculating $\sum_{l=0}^{\infty}\binom{l+100}{l}0.5^l 0.5^{100}$ and $\sum_{l=0}^{\infty}l \binom{l+100}{l}0.5^l 0.5^{100}$

Is there any formula for calculating $\sum_{l=0}^{\infty}\binom{l+100}{l}0.5^l 0.5^{100}$ and $\sum_{l=0}^{\infty}l \binom{l+100}{l}0.5^l 0.5^{100}$?
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### Find the probability of getting one diamond and one spade in a five-card hand, using binomial coefficients.

A five card hand is dealt at random from a standard $52$ card deck. Let $X = \text{# spades}$ and $Y = \text{# diamonds}$. Find $P(X = 1\text{ and }Y =1)$. Leave your answer as a ratio of products of ...
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### Alternating series of compositions of triangular numbers

I'm modeling a process which involves a subset $S$ of a large number $n_A$ of objects - call them balls. Each time I add a ball to $S$, it may dislodge another ball with probability proportional to ...
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### Closed form for binomial sum with absolute value

Do you know whether the following expression has a (nice) closed form or a close enough approximation? $$\frac{1}{2^n}\sum_{k=0}^{n} \binom{n}{k}|n-2k|$$ Thanks a lot :) Cheers, M.
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### Double counting proof of binomial problem

The assignment is to prove the following assertion using the method of double counting and explaining which pairs were counted. $$\dbinom{n+1}{k+1} = \sum_{i = k}^{n} \dbinom{i}{k}$$ Left side is ...
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### Finding the coefficient of expansion

Question: Find the coefficient of $x^{11}$ in the expansion of:$$(1+x^2)^4(1+x^3)^7(1+x^4)^{12}$$ The traditional way of doing this, as far as I know, is to first find the coefficient of each term ...
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### Of Balls in Bins in Different Sections with Caps

Problem: There are $19$ bins: $7, 5, 7$ in the left, centre and right sections respectively. There are $8$ balls, some or all of which are to be put into these bins with the following ...
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### Combinatorial identity / expected distance of random walk

I am struggling to verify the following identity. $$\binom{2m}{m} \frac{m}{2} = \sum_{j=1}^m j \binom{2m}{m+j}$$ I've tried induction, but I run into issues inside the sum. I can't see a ...
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### Error of Stirling’s approximation for Binomial with central limit theorem

So the question asks: Let $X_n$~Bin(2n,1/2),use Stirling’s approximation for $n!$ to show $P [X_n = n]$~ $1/√(πn)$ as $n→ ∞$, and show the error in the estimate for $P [X_n ≤ n]$, given by the central ...
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### Computing the coefficient of $x^n$ in the following expansion

The coefficient of $x^{-n}$ in the expansion of $\frac{2-3x}{1-3x+2x^2}$ is $a.)$ $(-3)^n - (2)^{\frac{1}{2}n -1}$ $b.)$ $2^n + 1$ $c.)$ $3(2)^{\frac{1}{2}n - 1} - 2(3)^n$ $d.)$ None of the ...
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### Closed form of a finite sum with binomial coefficients

In general, has $\sum_{k=a}^{b} \binom{n}{k}$ a closed form (with $0\le a\le b\le n$)?
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### Check whether or not a triangular number is triangular is the square-sum of two other consecutive triangular numbers

I'm trying to write a program that would tell me whether or not a triangular number, a number of the form $\frac{(n)(n+1)}{2}$ is the sum of the squares of two other consecutive triangular numbers. It ...
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### Coefficient of Multinomial kind of expression

How do I find the Multinomial coefficient of expression. For example $(x+y+z+w+6)^8$ let say I want the coefficient of xyzw. I know the answer in the simple case of $(x+3)^5$ , for $x^2$ it will ...
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### Is there a name for this combinatorial identity?

I found this identity in a textbook that I own but they did not name the identity and I had some trouble finding it online. Does anyone know the name of the identity and if I can find a resource about ...
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### Proving a binomial coefficient identity [duplicate]

I'm having some trouble with the following proof: $$\sum^k_{a=0} {{n}\choose{a}}{{m}\choose{k-a}} = {{n+m}\choose{k}}$$ I'm trying to prove this to learn a couple of things about the Pascal's ...
### How in this world can I simplify this $\sqrt 2\cdot(1/(\sqrt2)-1/(\sqrt2)\cdot i)^{31}$ ????
I have a problem, obviously. I am doing some maths and now I have to simplify this: $\sqrt 2\cdot(1/(\sqrt2)-1/(\sqrt2)\cdot i)^{31}$ ????. But I just don´t know how ???? I´ve started simplifying by ...
I was currently solving a question of permutations and in that I had to find the total ways of something. The answer was ${8\choose 4}$ which has last digit $0$ . A random thought that came to my ...