Given that we need to prove .

Consider L.H.S:

We know that ⁿC_r + ⁿC_{r + 1} = ^{n + 1}C_{r + 1}

⇒ ²ⁿC_n + ²ⁿC_{n – 1} = ^{2n + 1}C_n

We know that

And also n! = n(n – 1)(n – 2)…………2.1

⇒

⇒

⇒

⇒

⇒

⇒

⇒

= R.H.S

∴ L.H.S = R.H.S, thus proved.