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JEE Main · 2021 · 31 Aug 2021 · Shift 2 · Binomial Theorem

If the coefficient of

a^{7} b^{8}

in the expansion of

(a + 2 b + 4 ab)^{10}

is

K \cdot 2^{16}

, then

K

is equal to ____.

Answer: 315

Solution

By the multinomial theorem a term of

(a + 2 b + 4 ab)^{10}

is

\frac{10 !}{i ! j ! k !} a^{i} (2 b)^{j} (4 ab)^{k}

, carrying

a^{i + k} b^{j + k}

. Matching

a^{7} b^{8}

needs

i + k = 7

,

j + k = 8

with

i + j + k = 10

, giving

k = 5

,

i = 2

,

j = 3

. The coefficient is

\frac{10 !}{2 ! 3 ! 5 !} 2^{3} 4^{5} = 2520 \cdot 2^{13} = 315 \cdot 2^{16}

. Hence

K = 315

.

JEE Main · 2021 · 31 Aug 2021 · Shift 2 · Integer answer

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