First slide

Multinomial expansion

Question

The coefficient of x⁷ in ${(1 + 3 x - 2 x^{3})}^{10}$ is equal to

Moderate

A

62640
B

26240
C

64620
D

None of these

Solution

The coefficient of $x^{7} in {(1 + 3 x - 2 x^{3})}^{10}$
$= \frac{\sum 10!}{n_{1}! n_{2}! n_{3}!} (1)^{n_{1}} (3)^{n_{2}} (- 2)^{n_{3}}$
where $n_{1} + n_{2} + n_{3} = 10, n_{2} + 3 n_{3} = 7$
Different possibilities are as follow
$\begin{array}{ccc} n_{1} & n_{2} & n_{3} \\ 3 & 7 & 0 \\ 5 & 4 & 1 \\ 7 & 1 & 2 \end{array}$
$∴ Coefficient of x^{7} = \frac{10!}{3! 7!} (1)^{3} (3)^{7} (- 2)^{0} + \frac{10!}{5! 4! 1!} (1)^{5} (3)^{4} (- 2)^{1} + \frac{10!}{7! 1! 2!} (1)^{7} (3)^{1} (- 2)^{2}$
=62640

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