Question

prove that nCr+nCr-1= n+1Cr

Answer 1

nCr+nCr-1=n+1Cr
or LHS=n!/[r!(n-r)!]+n!/[(r-1)!(n-r+1)!]
=n!/[r(r-1)!(n-r)!]+ n!/[(r-1)!(n-r+1)(n-r)!]
=n!/[(r-1)!(n-r)!]{[1/r]+[1/(n-r+1)]}

=n!/[(r-1)!(n-r)!]{(n+1)/r(n-r+1)}
=(n+1)n!/[r(r-1)!(n-r+1)(n-r)!]
=(n+1)!/[r!(n-r+1)!]
=n+1Cr
i think it may be hope ful

Answer 2

LHS=n!/[r!(n-r)!]+n!/[(r-1)!(n-r+1)!]
=n!/[r(r-1)!(n-r)!]+ n!/[(r-1)!(n-r+1)(n-r)!]
=n!/[(r-1)!(n-r)!]{[1/r]+[1/(n-r+1)]}

=n!/[(r-1)!(n-r)!]{(n+1)/r(n-r+1)}
=(n+1)n!/[r(r-1)!(n-r+1)(n-r)!]
=(n+1)!/[r!(n-r+1)!]
=n+1Cr