The coefficient of x13 in the expansion of (1 – x)5 (1 + x + x2 + x

1.	The coefficient of x13 in the expansion of (1 – x)5 (1 + x + x2 + x3) 4 is(a) 4 (b) –4 (c) 0 (d) none of these
Answer» Correct option (a) 4 Explanation: Coefficient of x¹³in = (1 – x)⁵ (1 + x + x²+ x³) 4 = (1 – x)⁵ ((1 + x)(1 + x²))⁴ = (1 – x) {(1 – x)(1 + x)(1 + x² )}4 = (1 – x) {(1 – x⁴)⁴ = (1 – x) (1 – 4x⁴+ 6x⁸– 4x¹²+ x¹⁶) ∴ coefficient of x¹³ is –1 x – 4 = 4

The coefficient of x13 in the expansion of (1 – x)5 (1 + x + x2 + x3) 4 is(a) 4 (b) –4 (c) 0 (d) none of these

Answer»

Correct option (a) 4

Explanation:

Coefficient of x¹³in = (1 – x)⁵ (1 + x + x²+ x³) 4 = (1 – x)⁵ ((1 + x)(1 + x²))⁴

= (1 – x) {(1 – x)(1 + x)(1 + x² )}4

= (1 – x) {(1 – x⁴)⁴

= (1 – x) (1 – 4x⁴+ 6x⁸– 4x¹²+ x¹⁶)

∴ coefficient of x¹³ is –1 x – 4 = 4