# Tagged Questions

Questions on the factorial function, $n!=n\times(n-1)\times\cdots\times1$. Consider using the tag (gamma-function) if dealing with noninteger arguments.

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### What is the name of the identity: (2a)!=(a!*Γ(a-1/2) 4^a)/√π?

I just derived'(2a)!=(a!*Γ(a-1/2) 4^a)/√π'. My teacher has asked me to do some research over it. So my first question is. What is the name of this identity? Could it be a pi identity (if that is ...
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### Prove that $n^{n+1} \leq (n+1)^{n} \sqrt[n]{n!}$

Let $n$ be a positive integer. I conjectured that the following inequality is true $$n^{n+1} \leq (n+1)^{n} \sqrt[n]{n!} .$$ Anyhow I could neither prove nor disprove it. I ...
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### Find $a,b,c$ such that $a!\times b!\times c!=d!$ [duplicate]

I have to find $a,b,c \in \mathbb{N}$ such that-$a!\times b!\times c!=d!$ Answer given in my book is $3!\times5!\times7!=10!$(But it is written that other answers are also possible). What is a ...
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I'm trying to improve my knowledge of statistics and develop my intuition for solving statistical problems. While doing so I've worked on the following exercise: There are 20 players in a checkers ...
Verify the following identity algebraically (writing out the binomial coefficients as factorials).$${n \choose k}{k \choose m} = {n \choose m}{n-m \choose k-m}$$ So far, these are my steps: $$\frac{... 1answer 29 views ### Probability of Bridge Hands Using Distributions In a bridge deal, what is the probability that: a) West has five spades, two hearts, three diamonds, and three clubs? b) North and South have five spades, West has two spades, and East has one spade? ... 1answer 39 views ### One Dimensional Random Path Walker Problem The Probabilities involving 3 equally possible moves in 1D line. Imagine a one-dimensional line with a "walker" in the middle position (x=0) Walker can make one of the following moves ... 2answers 350 views ### I am stuck on proving \frac1{2!}+\frac2{3!}+\dots+\frac{n}{(n+1)!}=1-\frac1{(n+1)!} by induction, could anyone check my work? I will skip the Base Case step. This is the questions. Use mathematical induction to prove that$$\frac{1}{2!}+\frac{2}{3!}+\cdots+\frac{n}{(n+1)!}=1-\frac{1}{(n+1)!}$$for all integers n\ge 1. ... 2answers 80 views ### Limit of a function including factorials$$ \lim \frac{(2x)!}{(x! \cdot 2^x)^2} $$How can I deal with problems including factorials as the same as this problem 2answers 36 views ### What value of c makes this true? Since \lim_{x \rightarrow \infty}\frac{(x)!}{x^{x}} = 0 and \lim_{x \rightarrow \infty}\frac{(2x)!}{x^{x}} = \infty Is there a value c (or range of values) where \lim_{x \rightarrow \infty}\... 1answer 23 views ### Is it a correct equation for permutations with sets of indistinguishable objects? C(n, r) = P(n, r)!/r! = n!/r!ㆍ(n-r)! I'll check if the right hand side of the above equation in Theorem 9.5.2 is correct by expanding the left hand side. C(n, n_1)ㆍC(n-n_1, n_2)ㆍC(n-n_1-n_2, n_3)\... 2answers 43 views ### limit of fraction with factorials I am trying to take the limit of the following fraction :$$ \lim_{N \to\infty} \frac { N !}{(N-r)!}  Attempts : I tried using the Stirling approximation $\ln(n!) =n \ln n - n$ but I figured it ...
I've recently been learning factorials in school. If there is an equation (in $\mathbb N$) with $(n-5)!$, I have to ensure that $n$ is not 1, 2, 3 or 4. I've been told that I should write domain: \$D =...