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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Variance in offspring genotypes

Background Here is first some vocabulary: Diploid: phase in the life cycle where the individuals carry two chromosomes of each type, just like in humans (exception of the sexual chromosomes). ...
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Problem proving $ \sum_{r=0}^{n-1} \binom{2n-1}{r} = 2^{2n-2} $

I'm stuck at proving the following. $$ \sum_{r=0}^{n-1} \binom{2n-1}{r} = 2^{2n-2} $$ I know that I have to use the Binomial theorem like this, letting x=1,y=n in $(x+y)^{2n-1}$ $$ ...
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Prove $\sum_{i=0}^{i=x} {x \choose i} {y+i \choose x}+\sum_{i=0}^{i=x} {x \choose i} {y+1+i \choose x}$

How to prove that $$\sum_{i=0}^{i=x} {x \choose i} {y+i \choose x}+\sum_{i=0}^{i=x} {x \choose i} {y+1+i \choose x}=\sum_{i=0}^{i=x+1} {x+1 \choose i} {y+i \choose x}$$ ? I tried to break the right ...
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Proving $\sum_i \binom{k}{k-i} \binom{n-k}{i} = \binom{n}{k}$

I am in the middle of a probability question. The question is indeed simple. For the sake of clarity of the notation, I also include the question here, which is from Sheldon M. Ross's Introduction to ...
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Sums of binomial coefficients

Does anyone know something about the following sums? $$ S_m(n)=\sum\limits_{k=o}^n(-1)^k{mn\choose mk} $$ Notice that $S_m(n)=0$ for odd $n$, so we only consider $S_m(2n)$. It holds that $S_0(2n)=1$, ...
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1answer
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Placing m books on n shelves

If we let m and n be integers with $m \ge n \ge 1$. how many ways are there to place m books on n shelves, if there must be at least one book on each shelf? the order matter. How do I solve this, do I ...
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Solve Identity about Combination

Find the values of a and b such that $\binom{2n}{2} = a\binom{n}{2} + b(n^2)$ This is a past year question about Introduction of Combinatorics in my university.
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2answers
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How many ways can the school choose a President Vice President?

There are n >= 4 students. The school has a Board of Directors, consisting of one president and three vice-presidents. The entire board consists of four distinct students. How can I prove that ...
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4answers
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Question about ${n+1\choose k} = {n\choose k} + {n\choose k-1}$ proof?

I've found past proofs of this problem and for the most part I'm able to follow. $$\eqalign{{n\choose k}+{n\choose k-1}&= {n!\over (n-k)!k!}+ {n!\over (n-(k-1))! (k-1)!} \text{ (step 1)}\cr ...
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The sum of binomial coefficients up to $k\le n/4$ does not exceed the $k$th coefficient

How would you prove the following (for when $k\leq\frac{n}{4}$)? $$\sum_{i=0}^{k-1} \binom ni \le \binom nk$$
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1answer
23 views

A urine test, the VMA test

Neuroblastoma is a rare, serious, but treatable disease. A urine test, the VMA test, has been developed that gives a positive diagnosis in about 70 % of cases of neuroblastoma. It has been proposed ...
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1answer
29 views

Binomial coefficients and order of infinity

Which among $$ \left(2\,k+1 \atop j\right),~~j=1,3,5,...,2\,k+1 $$ has the larger order of infinity when $k\rightarrow\infty$? I am pretty sure that the largest order is reached around $j=k$ but I ...
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1answer
42 views

What is the coefficient of $x^4y^3z^3$ in the expansion of $(5x+1y+5z)^{10}$?

What is the coefficient of $x^4y^3z^3$ in the expansion of $(5x+1y+5z)^{10}$? So, would I start by using the binomial or multinomial theorem? Not entirely sure where to start here?
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What is the coefficient of$ x^6y^1$ in the expansion of $(3x^2+y)^4$?

So this is what I have so far. $(3x^2)^4$ + $\binom{4}{1}(3x^2)^3(y)$ Why is the answer not 4? How do I continue?
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1answer
25 views

A drug treatment [on hold]

A certain drug treatment cures 90 % of cases of hookworm in children. Suppose that 20 children suffering from hookworm are to be treated, and that the children can be regarded as a random sample from ...
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1answer
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Show by committee selection argument

First post in Stack Exchange and feel bad to be in need of help. But, I'm having a hard time understanding this one or rather showing the argument. $\binom{n}{k} = \binom{n-2}{k-2} + ...
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1answer
28 views

What is the combinatorial interpretation of the product of binomial coefficients?

Full disclosure: This question is relating to a homework question. It's not a homework question itself, but rather a clarifying question to help myself get a handle on the actual question. Suppose I ...
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Calculate the binomial sum $ I_n=\sum_{i=0}^n (-1)^i { 2n+1-i \choose i} $

I need any hint with calculating of the sum $$ I_n=\sum_{i=0}^n (-1)^i { 2n+1-i \choose i}. $$ Maple give the strange unsimplified result $$ I_n={\frac {1/12\,i\sqrt {3} \left( - \left( \left( ...
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1answer
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Sum of products of binomial coefficients is equal to another binomial coefficient [duplicate]

Need help in proving (by induction or by combinatorics) the following statement Is it possible to do it by induction? there are 3 veriables and I think I cannot easily do it by induction. Correct? ...
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1answer
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Combinatorial Analysis: Fermat's Combinatorial Identity

I was looking through practice questions and need some guidance/assistance in Fermat's combinatorial identity. I read through this on the stack exchange, but the question was modified in the latest ...
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2answers
41 views

Sum of Binomial Coefficient products

I am trying to prove that $$\sum\limits_{y=0}^d \frac{{2x \choose y} {2d-2x \choose d-y} }{2d \choose d} = x $$ So far, I have tried using induction on $d$ but I am having trouble using the ...
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46 views

Prove combination identity

$ \sum_{k=0}^n {2k \choose k} {2n-2k \choose n-k} = 4^n $ I tried with mathematical induction only to fail. Is this formula related to some special function like Beta, Gamma, etc?
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Probability for a 'pair' to occur when rolling 5 dice

5 fair dice are rolled. A pair is defined to be any number that shows up twice, while the rest of the dice show different numbers (to the number on the pair and to each other). I am looking for the ...
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Gosper's Identity $\sum_{k=0}^n{n+k\choose k}[x^{n+1}(1-x)^k+(1-x)^{n+1}x^k]=1 $

The page on Binomial Sums in Wolfram Mathworld http://mathworld.wolfram.com/BinomialSums.html (Equation 69) gives this neat-looking identity due to Gosper (1972): $$\sum_{k=0}^n{n+k\choose ...
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1answer
26 views

binominal inequality - checking

I have to show that $\displaystyle \binom{n}{k}<\binom{n}{l}$ for $\displaystyle 0 \le k <l \le\frac{n}{2}$ where $n,k,l$ are an integers. I think I solved it but I'm not sure if my approach is ...
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133 views

Inequality $\binom{2n}{n}\leq 4^n$

I would like to prove the following inequality, for $n=0,1,2,...$, $$ \binom{2n}{n}\leq 4^n.$$ I already proved it by induction, and I'm looking for another proof.
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Transformation of a sum

I want to prove the following or a similar result: For $1\le k \le n$ \begin{align}&1-\sum\limits_{j=k+1}^n\binom nj(1-x)^jx^{n-j}~~~~~~(1)\\ ...
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1answer
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Function related to Harmonic numbers, the Pascal triangle, Logarithmic integral and the Polylogarithm.

What function satisfies the following: Let the matrix: $$\displaystyle T = \left(\begin{matrix} 1&0&0&0&0&0&0&\cdots \\ 1&1&0&0&0&0&0 \\ ...
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Solve for $n$, where $n$ is a positive integer

I have $$ {n \choose 2} = 21 $$ and as the title mentions I have to solve for $n$, but so far all I have managed to get to is $$n^2 -n =42 $$ and from there I'm completely lost. Any hints would ...
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3answers
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Show that $ \sum_{k=0}^{r} \binom{r-k}{m} \binom{s+k}{n} = \binom{r+s+1}{m+n+1} $?

I can't resolve this exercise and I need a tip. $$ \sum_{k=0}^{r} \binom{r-k}{m} \binom{s+k}{n} = \binom{r+s+1}{m+n+1} $$ where $ n \geq s $.
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1answer
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Finding the coefficient of $x^{46}$ in an expression

In this problem, I found that the answer is the coefficient of $x^{46}$ in $\left(\displaystyle\sum_{r=0}^{3} x^r \right)^6\left(\displaystyle\sum_{r=0}^{8} x^r \right)^4$ Is there possibly a way to ...
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Find $a_1$ given that $(1+x)^{100} = \sum_{i=0}^{100} a_ix^i$

If $(1+x)^{100} = \sum_{i=0}^{100} a_ix^i$, then $a_1$ is .. The options are $1$, $2$, $99$ or $100$. I'm sure the problem is trivial, but I just don't understand what is meant.
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Challenge: How to prove this identity between bi- and trinomial coefficients?

This question is the continuation of its predecessor. Using the convention that trinomial coefficients $$ \binom{n}{k_1,k_2,k_3}=\frac{n!}{k_1! k_2! k_3!} $$ are zero if $k_i<0$ or $\sum_i k_i\neq ...
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Evaluate $ \sum\limits_{j \in \mathbb{Z}_{\geq 0}} {n \choose r+kj}$ where $n,k$ are fixed

Is there a general way/technique to evaluate $ \sum\limits_{j \in \mathbb{Z}_{\geq 0}} {n \choose r+kj}$ in terms of $r$, where we consider $n$ and $k$ fixed natural numbers and $n > k$? (here, ...
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Proof of Pascal' identity

The identity $$\binom{x+1}{k}-\binom{x}{k}=\binom{x}{k-1}$$ is claimed to hold (using the binomial polynomials, considered as lying in $\mathbf{Q}[x]$) for $k$ at least $1$. Proof: by the usual ...
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1answer
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Consecutive terms in Pascal's Triangle

is it known whether or not there are infinitely many pairs of consecutive terms in this sequence: http://oeis.org/A006987 ? The sequence is the list of numbers expressible in the form ...
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1answer
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$\sum_{k+M = 0}^n {n \choose k} {n-k \choose m} = 3^n$ help with combinatorial reasoning

$\sum_{k+M = 0}^n {n \choose k} {n-k \choose m} = 3^n $ I have worked the cases for $n=2$, $n=3$, and $n=4$ by hand and it appears to be true. $n=2$: $${2\choose 0}{2\choose 0} + {2\choose ...
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Algorithm for determining whether certain numbers appear in Pascal's triangle

Is there any easy characterisation for the numbers which appear in Pascal's triangle that ARE NOT $\dbinom{n}{1}$, $\dbinom{n}{n-1}$? Is there a fast way to determine if some number (given its prime ...
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Simplifying a sum with factorials and permutation counts

I am able to simplify it to a form of summation $$\frac{{n\choose r}}{K {a\choose b}}$$ but not able to proceed any further.
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Why is $\sum\limits_{b=1}^{t-1} {t \choose b} 2^{t-b} = (3^t - 2^t - 1)$

Why is $$\sum\limits_{b=1}^{t-1} {t \choose b} 2^{t-b} = (3^t - 2^t - 1)$$ Thanks.
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3answers
141 views

Using Binomial Theorem to prove the following [duplicate]

$$\large\sum_{j=0}^n (-1)^j {n\choose j}={n\choose 0}-{n\choose 1}+.....+\pm{n\choose n}=0 $$ I'm confused by the last part of the equation $\pm$. it seems imply that the sum would be equal to 0 no ...
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2answers
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Factorial as a sum. Insight appreciated

I recently posted an answer to a question about ways to express the factorial function as a sum. I posted the following formula, which I discovered several years ago and I haven't seen anywhere else: ...
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1answer
38 views

Find coefficient of $x^3$ in (2+x) ^(3/2)/(1-x)

I can expand $\dfrac{(2+x)^{3/2}}{1-x}=(1+x+x^2+\ldots)\left({3/2\choose0}+{3/2\choose1}(x+1)+{3/2\choose2}(x+1)^2+\ldots\right)$, but that doesn't seem to lead anywhere.
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How to prove the following limit: $\lim_{n \to +\infty} 4^n\left[\sum_{k=0}^n (-1)^k{n\choose k}\ln (n+k)\right]=0$?

How to prove the following limit: $$\lim_{n \to +\infty} 4^n\left[\sum_{k=0}^n (-1)^k{n\choose k}\ln (n+k)\right]=0?$$
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Generating functions to solve number of integer solution problem

If I have $x_1 + x_2 + x_3 =10$ with $1\leq x_1 \leq 5, \; 2 \leq x_2 \leq 6, \;3 \leq x_3 \leq 9$ I know that I compute $(t^1+\dots + t^5)(t^2 +\dots + t^6)(t^3+\dots +t^9)$ and look at the ...
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1answer
34 views

Meaning of coefficients when multiplying together $(1+x^a)(1+x^b)(1+x^c)\cdots$?

I know that when you multiply out $(1+x)^n$, the coefficient of $x^a$ tells you how many ways you can pick a of the brackets to use the x from to make $x^a$. I was wondering whether there is a meaning ...
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3answers
101 views

Evaluate the sum

I need to evaluate the following sum, which depends on $n \in \mathbb N$ (call it $S(n)$ if you will) $$ \sum_{i=0}^{n} (-1)^{n-i} \binom{n}{i} f(i)$$ where $f$ defined over $\mathbb N$ is ...
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1answer
48 views

Number after binomial coefficient

What does the number 2 mean in this picture? link Thanks. Sorry I'm not a math expert. I need this to make a program in C to generate a trinomial triangle for a guy who asked it on StackOverflow ...
6
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2answers
478 views

How to solve 0.5 choose 4?

I was solving this problem for homework. It says, in the problem, that if n is positive you use the generalized definition of binomial coefficients. In my case, n is positive so I just plugged n= 0.5 ...
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1answer
53 views

How prove this sum $\sum\limits_{n=k}^{\infty}\dbinom{n}{k}\left(\dfrac{-z}{1-z}\right)^n$

prove or disprove $$\sum_{n=k}^{\infty}\binom{n}{k}\left(\dfrac{-z}{1-z}\right)^n= (1-z)(-z)^k$$ my try: since ...