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Question
Find a, if the coefficients of x2 and x3 in the expansion of (3 + ax)9 are equal.
Solution
\[(3 + ax )^9 \]
\[ =^{9}{}{C}_0 . 3^9 . (ax )^0 + ^{9}{}{C}_1 . 3^8 . (ax )^1 + ^{9}{}{C}_2 . 3^7 . (ax )^2 + ^{9}{}{C}_3 . 3^6 . (ax )^3 + . . .\]
We have Coefficient of x2 = Coefficient of x3
\[^{9}{}{C}_2 \times 3^7 a^2 = ^{9}{}{C}_3 \times 3^6 a^3 \]
\[ \Rightarrow a = \frac{^{9}{}{C}_2}{^{9}{}{C}_3} \times 3\]
\[ = \frac{9! \times 3! \times 6! \times 3}{2! \times 7! \times 9!}\]
\[ = \frac{9}{7}\]
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