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The coefficient of $a^8 b^4 c^9 d^9$ in $(a b c+a b d+a c d+b c d)^{10}$ is
Option: 1
10!

Option: 2
$\frac{10 !}{8 ! 4 ! 9 ! 9 !}$

Option: 3
2520

Option: 4
None of these

Answers (1)

General term in $(a b c+a b d+a c d+b c d)^{10}$ is given by

$\begin{aligned} & \frac{10 !}{p ! q ! r ! s !}(a b c)^p(a b d)^q(a c d)^r(b c d)^s \text {, where } p+q+r+s=10 \\ & =\frac{10 !}{p ! q ! r ! s !} a^{p+q+r} b^{p+q+s} c^{p+r+s} d^{q+r+s} \end{aligned}$

To find the coefficient of $a^8 b^4 c^9 d^9$ , we have p+q+r=8,

p+q+s=4, p+r+s=9, q+r+s=9

Also, p+q+r+s=10
On solving, we get p=1, q=1, r=6, s=2
$\therefore \quad \text { Coefficient of } a^8 b^4 c^9 d^9=\frac{10 !}{1 ! 6 ! 2 !}=2520$

Posted by

Ramraj Saini

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