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How to prove $\sum_{r=0}^{n}\left(-1\right)^r\binom{n}{r}\left(n-r\right)^n=n!$

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#summation #solution-verification #induction #factorial
3.1k
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Evaluating $\sum\limits_{x=2}^\infty \frac{1}{!x}$ in exact form.

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#calculus #sequences-and-series #special-functions #factorial #gamma-function
73
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Equality involving combinatorics

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#combinatorics #factorial
46
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Why do the rounded solutions of $n^x=x!$ follow this $2, 3, 3$ pattern as $n$ increases?

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#exponentiation #factorial
197
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Divergent sum of $\sum_{n=0}^\infty (-1)^n \left(\alpha n\right)!$

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#sequences-and-series #factorial #divergent-series
99
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Showing that the partial sums of this alternating series with factorials do not vanish before limit

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#sequences-and-series #summation #factorial
244
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Is $\log(n!)=o(\log(n^n))$?

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#algorithms #logarithms #asymptotics #factorial
200
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The Product of two rising factorials

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#special-functions #factorial #pochhammer-symbol
186
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How to say $!n$ out loud

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#soft-question #factorial #derangements #pronunciation
43
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Can this ‘function’, similar to a factorial but divides numbers instead, be useful?

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#factorial
468
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Find the last $2$ digits of the value of $(1! + 2! + 3! + 4! + .... + 2016!)^{42}$.

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#elementary-number-theory #modular-arithmetic #contest-math #exponentiation #factorial
381
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Factorial-like product where the factors are offset

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#sequences-and-series #numerical-methods #approximation #factorial #gamma-function
87
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Factorials and Place Value

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#elementary-number-theory #factorial #prime-factorization #multiplicative-function
122
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Simplifying $\sum_x\frac{(A+2x)!}{(2x)!}$

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#summation #factorial
325
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Algebraic proof regarding falling factorials

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#combinatorics #factorial

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