Question

The coefficient of x 101 in the expression (5 + x)500 + x(5 + x)499 + x 2(5 + x)498 + ……+ x 500, x > 0, is

• A. 501 C 101 (5)399
• B. 501 C 101 (5)400
• C. 501 C 100 (5)400
• D. 500 C 101 (5)399

Answer 1

Answer: (A)

501 C 101 (5)399

Answer 2

The correct option is(A): 501 C 101 (5)399

Coeff. of x101 in $\frac{x_{500}[(\frac{x+5}{x})^{501}-1]}{\frac{x+5}{x}-1}$

= Coeff. of x101 in $\frac{1}{5}[(x+5)^{501-x^{501}}]$

$=\frac{1}{5}^{501}c_{101}.5^{400}$

$=501C_{101}.5^{399}$