Question

Show that n+1C = nCr-1 + nr.

Answer 1

Answer: The proof is given below.

Step-by-step explanation: We are given to show that the following equality is true :

`^(n+1)C_r=^nC_(r-1)+^nC_r.`

We know that

the number of combinations of n different things taken r at a time is given by

`^nC_r=(n!)/(r!(n-r)!).`

Therefore, we have

`R.H.S.\\\\=^nC_(r-1)+^nC_r\\\\\\=(n!)/((r-1)!(n-(r-1))!)+(n!)/((r)!(n-r)!)\\\\\\=(n!)/((r-1)!(n-r+1)!)+(n!)/((r)!(n-r)!)\\\\\\=(n!)/((r-1)!(n-r+1)(n-r)!)+(n!)/(r(r-1)!(n-r)!)\\\\\\=(n!)/((r-1)!(n-r)!)\left((1)/(n-r+1)+(1)/(r)\right)\\\\\\=(n!)/((r-1)!(n-r)!)\left((r+n-r+1)/((n-r+1)r)\right)\\\\\\=(n!)/((r-1)!(n-r)!)*(n+1)/((n-r+1)r)\\\\\\=((n+1)!)/(r!(n-r+1)!)\\\\\\=((n+1)!)/(r!((n+1)-r)!)\\\\\\=^(n+1)C_r\\\\=L.H.S.`

Thus,
`^(n+1)C_r=^nC_(r-1)+^nC_r.`

Hence proved.