The coefficient of x5 in the expansion of (1 + x)21 + (1 + x)22 +...

1.	The coefficient of x5 in the expansion of (1 + x)21 + (1 + x)22 +.........+(1 + x)30 is(a) 31 C5 – 21C5(b) 31C6 – 21C6 (c) 30C6 – 20C6 (d) None of these
Answer» Correct option (b) ³¹C₆ – ²¹C₆ Explanation: Co-efficient of x⁵ in (1 + x)²¹+ (1 + x)²²+............+ (1 + x)³⁰ = Co-efficient of x⁵ in (1 + x)²¹{(1 + x)¹⁰ -1}/(1 + x) - 1 coefficient of x⁶ in (1 + x)³¹ – (1 + x)²¹ is ³¹C₆ - ²¹C₆

The coefficient of x5 in the expansion of (1 + x)21 + (1 + x)22 +.........+(1 + x)30 is(a) 31 C5 – 21C5(b) 31C6 – 21C6 (c) 30C6 – 20C6 (d) None of these

Answer»

Correct option (b) ³¹C₆ – ²¹C₆

Explanation:

Co-efficient of x⁵ in (1 + x)²¹+ (1 + x)²²+............+ (1 + x)³⁰

= Co-efficient of x⁵ in (1 + x)²¹{(1 + x)¹⁰ -1}/(1 + x) - 1

coefficient of x⁶ in (1 + x)³¹ – (1 + x)²¹ is ³¹C₆ - ²¹C₆