First slide

Multinomial expansion

Question

The coefficient of $a^{8} b^{4} c^{9} d^{9}$ in $(abc + abd + acd + bcd)^{10}$ is

Moderate

A

10!
B

$\frac{10!}{8! 4! 9! 9!}$
C

2520
D

$5040$

Solution

$a^{10} b^{10} c^{10} d^{10} {(\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d})}^{10}$
General term= $\frac{10!}{p_{1}! p_{2}! p_{3}! p_{4}!} {(\frac{1}{a})}^{p_{1}} {(\frac{1}{b})}^{p_{2}} {(\frac{1}{c})}^{p_{3}} {(\frac{1}{d})}^{p_{4}} w h e r e p_{1} + p_{2} + p_{3} + p_{4} = 10$
Therefore the required coefficient is equal to the coefficient of
$a^{- 2} b^{- 6} c^{- 1} d^{- 1} in {(\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d})}^{10}$ , which is given by
$\frac{10!}{2! 6! 1! 1!} = \frac{10 \times 9 \times 8 \times 7}{2} = 2520$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App