If-1-x-n-C-0-C-1-x-C-2-x-2-C-n-x-n-then-C-1-2-C-2-2-C-n-2-is-equal-to-

Question Number 18762 by 786786AM last updated on 29/Jul/17

If {(1 + x)}^{n} = C_{0} + C_{1} x + C_{2} x^{2} + \dots + C_{n} x^{n},

then C_{1}^{2} + C_{2}^{2} + \dots . + C_{n}^{2} is equal to

Answered by Rishabh#1 last updated on 30/Jul/17

{(1 + x)}^{2 n} = {(1 + x)}^{n} {(x + 1)}^{n}

= (^{n} C_{0} x^{0} +^{n} C_{1} x + \dots +^{n} C_{n} x^{n}) \times

(^{n} C_{0} x^{n} +^{n} C_{1} x^{n - 1} + \dots +^{n} C_{n} x^{0})

c o e f f e c i e n t o f x^{n} i n l h s =^{2 n} C_{n}

c o e f f e c i e n t o f x^{n} i n r h s =

C_{0}^{2} + C_{1}^{2} + . . + C_{n}^{2}

h e n c e

C_{0}^{2} + C_{1}^{2} + . . + C_{n}^{2} =^{2 n} C_{n}