Sum of coefficients of terms which contains integralegral po... | MATH - PROPERTIES OF BINOMIAL COEFFICIENTS | SolveeIt | Solveeit

Today, August 24

Question

Sum of coefficients of terms which contains integral powers of x in the expansion (2 + $\sqrt{x}$ )³⁰ is-

A

$\frac{2^{30} + 1}{2}$

B

$\frac{2^{30} - 1}{2}$

C

$\frac{3^{30} - 1}{2}$

D

$\frac{3^{30} + 1}{2}$

Answer

$\frac{3^{30} + 1}{2}$

Explanation

Solution

(2 + $\sqrt{x}$ )³⁰

= ³⁰C₀2³⁰ + ³⁰C₁2²⁹ $\sqrt{x}$ + ³⁰C₂2²⁸( $\sqrt{x}$ )² + ..... ³⁰C₃₀( $\sqrt{x}$ )³⁰

S = ³⁰C₀2³⁰ + ³⁰C₂2²⁸ + ³⁰C₄2²⁶ + ..... + ³⁰C₃₀

(x + 1)³⁰ = ³⁰C₀ x³⁰ + ³⁰C₁ x²⁹ + .... ³⁰C₃₀ x⁰

put x = 2

3³⁰ = ³⁰C₀2³⁰ + ³⁰C₁2²⁹ + ....... + ³⁰C₃₀ x = –2

1 = ³⁰C₀2³⁰ – ³⁰C₁2²⁹ + ...... + ³⁰C₃₀

$\frac{3^{30} + 1}{2}$ = ³⁰C₀2³⁰ + ³⁰C₂ 2²⁸ + ...... + ³⁰C₃₀