Prove that nCr plus nCr minus 1 equals n plus 1Cr?

Answer 1

nCr + nCr-1 = n!/[r!(n-r)!] + n!/[(r-1)!(n-r+1)!]

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

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

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

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

= n+1Cr