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Relation between Coefficient and Indices of X and Y

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

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Solution

The correct option is C 12
$(1 + x)^{m} (1 - x)^{n} = (1 + m x + \frac{m (m - 1)}{2} x^{2} + . . .) \times (1 - n x + \frac{n (n - 1)}{2} x^{2} + . . .) ∴ C o e f f i c i e n t o f x = m - n = 3 (g i v e n) C o e f f i c i e n t o f x^{2} = \frac{n (n - 1)}{2} - n m + \frac{m (m - 1)}{2} = - 6 (g i v e n) \Rightarrow n^{2} - n - 2 m n + m^{2} - m = - 12 \Rightarrow (m - n)^{2} - (m + n) = - 12 \Rightarrow (3)^{2} - (m + n) = - 12 \Rightarrow m + n = 21 ∴ S o l v i n g m - n = 3, m + n = 21, w e g e t m = 12, n = 9$

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