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I am a beginner in math I was developing a coding problem and here is the pattern which I found: Help me understand the pattern.

Factorial of n will be the sum of all the product of all permutation of digits(1,n-1) taken from 1 to n-1 at a time, with the addition of 1.

I may not sound clear but below is the example. For Example:

factorial 4:
numbers = 1,2,3
factorial 4 = [(1)+(2)+(3)]+[(1x2)+(1x3)+(2X3)]+[(1x2x3)] = 23+1

factorial 5:
numbers = 1,2,3,4
factorial 5 = [(1)+(2)+(3)+(4)]+[(1x2)+(1x2)+(1X3)+(1x4)+(2+3)+(2+4)+(3+4)]+[(1x2x3)+(1x2x4)+(1x3x4)+(2x3x4)]+[(1x2x3x4)] = 119+1

Can we prove this mathematically?

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The terms that you supply are the coefficients of the development of the polynomial

$$(x+1)(x+2)(x+3)\cdots(x+n)$$

i.e. $$x^n+(1+2+3+\cdots n)x^{n-1}+(1\cdot2+1\cdot3+\cdots (n-1)n)x^{n-2}+\cdots+1\cdot2\cdots n.$$ See https://en.wikipedia.org/wiki/Vieta%27s_formulas, https://en.wikipedia.org/wiki/Elementary_symmetric_polynomial and https://en.wikipedia.org/wiki/Falling_and_rising_factorials.

Now, set $x=1$ to get the sum of the coefficients.


Bonus:

Setting $x=-1$ you obtain that the sum with alternating signs is $0$.

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