Something I can't seem to prove

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SUMMARY

The discussion centers on proving the combinatorial identity involving factorials: \(\frac{n!}{(n-i)! i!} + \frac{n!}{[n-(i+1)]! (i+1)!} = \frac{(n+1)!}{(n-i)! (i+1)!}\). The user struggles with simplifying the left-hand side, particularly with the term \([n - (i + 1)]!\). The key insight is recognizing that \([n - i] \cdot [n - (i + 1)]! = [n - i]!\), which is essential for combining the fractions. This identity is foundational in combinatorics and is used in generating Pascal's triangle.

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Skynt
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It should be really simple, but I think the algebra is bogging me down:

\frac{n!}{(n-i)! i!} + \frac{n!}{[n-(i+1)]! (i+1)!} = \frac{(n+1)!}{(n-i)! (i+1)!}

Can anyone show me the process of proving this? I don't see how the two expressions are equal...

essentially I can't get rid of the [n - (i + 1)]! term when I try to combine the two fractions and expand things.
 
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Notice that [n - i] * [n - (i + 1)]! = [n - i]!.
 
obviously *bangs head* thanks!
 
That is, by the way, the formula used for producing Pascal's triangle.
 

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