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If the coefficient of a7b8 in the expansion of a 2b 4ab... | null (Mathematics) | DivineJEE

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Mathematics

Exam

Mathematics

If the coefficient of a⁷b⁸ in the expansion of (a + 2b + 4ab)¹⁰ is K.2¹⁶, then K is equal to _____________.

Correct Answer:

315

Explanation

{{10!} \over {\alpha !\beta !\gamma !}}{a^\alpha }{(2b)^\beta }.{(4ab)^\gamma }

{{10!} \over {\alpha !\beta !\gamma !}}{a^{\alpha + \gamma }}.\,{b^{\beta + \gamma }}\,.\,{2^\beta }\,.\,{4^\gamma }

\alpha + \beta + \gamma = 10

..... (1)

\alpha + \gamma = 7

.... (2)

\beta + \gamma = 8

..... (3)

(2) + (3) - (1) \Rightarrow \gamma = 5

\alpha = 2

\beta = 3

so coefficients =

{{10!} \over {2!3!5!}}{2^3}{.2^{10}}

= {{10 \times 9 \times 8 \times 7 \times 6 \times 5} \over {2 \times 3 \times 2 \times 5!}} \times {2^{13}}

= 315 \times {2^{16}} \Rightarrow k = 315

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