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JEE Main · 2023 · 25 Jan 2023 · Shift 1 · Binomial Theorem

The constant term in the expansion of

(2 x + \frac{1}{x ^{7}} + 3 x^{2})^{5}

is ______.

Answer: 1080

Solution

The general term of the multinomial expansion is

\frac{5 !}{n _{1} ! n _{2} ! n _{3} !} (2 x)^{n_{1}} (x^{- 7})^{n_{2}} (3 x^{2})^{n_{3}}

with

n_{1} + n_{2} + n_{3} = 5

, whose power of

x

is

n_{1} - 7 n_{2} + 2 n_{3}

. Setting this to zero together with the count condition forces

n_{1} = 1, n_{2} = 1, n_{3} = 3

. The term is

\frac{5 !}{1 ! 1 ! 3 !} \cdot 2 \cdot 3^{3} = 20 \cdot 54 = 1080

.

JEE Main · 2023 · 25 Jan 2023 · Shift 1 · Integer answer

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