Question

Prove n!/k!(n-k+1)!+n!/k!(n-k)!=(n+1)!/k!(n+1-k)!

Answer 1

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

Answer 2

n!/k!*(n-k)! + n!/(k-1)!*(n-k+1)!
=n!/k(k-1)!(n-k)!+n!/(k-1)!(n-k+1)(n-k…
=[n!/(k-1)!(n-k)!]{1/k+1/(n-k+1)}
=[n!/(k-1)!(n-k)!][(n+1)/k(n-k+1)]
=(n+1)!/k!(n-k+1)!