Question

Prove that : n{n-1 C r-1}={n-r+1}{n C r-1}

Answer 1

Prove that : n{n-1 C r-1}={n-r+1}{n C r-1}

 

Let A = {n-1 C r-1}

Let B = n C r-1

 

A = (n-1)! /{(r-1)!(n-r)!}

B = n! / {(r-1)!(n-r+1)!}

 

Rearranging B,

B = n(n-1)! / {(r-1)!(n-r+1)(n-r)!}

B = n(n-1)! / {(r-1)!(n-r)!(n-r+1)}

B = {n/(n-r+1)} *(n-1)! / {(r-1)!(n-r)!}

B = {n/(n-r+1)} *A

An = (n-r+1)B   i.e.

n{n-1 C r-1}={n-r+1}{n C r-1}