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Greatest Binomial Coefficients

If in the exp...

Question

If in the expansion of $(1 + x)$ $^{m}$ $(1 - x)$ $^{n}$ , the coefficients of $x$ and $x^{2}$ are $3$ and $- 6$ respectively, then m is

6

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9

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12

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24

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Solution

The correct option is C $12$
$(1 + x)^{m} (1 - x)^{n} = (1 + m x + \frac{m (m - 1) x^{2}}{2!}) (1 - n x + \frac{n (n - 1)}{2!} x^{2})$
$= 1 + (m - n) x + [\frac{n^{2} - n}{2} - m n + \frac{(m^{2} - m)}{2}] x^{2}$
Given $m - n = 3$ or $n = m - 3$
Hence $\frac{n^{2} - n}{2} - m n + \frac{m^{2} - m}{2} = 6$
$\Rightarrow \frac{(m - 3) (m - 4)}{2} - m (m - 3) + \frac{m^{2} - m}{2} = - 6$
$\Rightarrow m^{2} - 7 m + 12 - 2 m^{2} + 6 m + m^{2} - m + 12 = 0$
$\Rightarrow - 2 m + 24 = 0 \Rightarrow m = 12.$

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