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Question

Coefficient of $$a^5 b^7 $$ in $$(a-2b)$$ $$^{12}$$ is


A
101376
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B
201356
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C
31064
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D
None
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Solution

The correct option is A $$-101376$$
$$(a - 2b)^{12} = [a + (-2b)]^{12}$$
General term $$T_{r+1} = C (12, r) a^{12-r}(-2b)^r$$
Putting $$12-r = 5$$ or $$12 -5 = r \Rightarrow r = 7$$
$$T_{7+1} = C (12, 7) a^{12 -7} (-2b)^7 = C (12, 7) a^5 (-2b)^7 = C(12, 7)(-2)^7 a^5 b^7$$
Hence required coefficient is $$C(12, 7) (-2)^7 = \displaystyle \frac{!2!}{7! 5!} \cdot 2^7 = \displaystyle \frac{-12 \times 11 \times 10 \times 9 \times 8 \times 7 !}{7! \times 5 \times 4\times 3 \times 2 \times 1} \times 2^7$$
$$ = -11 \times 9 \times 2^7 = -99 \times 128 = - 101376$$

Mathematics

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