Question:
Find a, if the coefficients of x2 and x3 in the expansion of (3 + ax)9 are equal.
Solution:
$(3+a x)^{9}$
$={ }^{9} C_{0} \cdot 3^{9} \cdot(a x)^{0}+{ }^{9} C_{1} \cdot 3^{8} \cdot(a x)^{1}+{ }^{9} C_{2} \cdot 3^{7} \cdot(a x)^{2}+{ }^{9} C_{3} \cdot 3^{6} \cdot(a x)^{3}+\ldots$
We have Coefficient of $x^{2}=$ Coefficient of $x^{3}$
${ }^{9} C_{2} \times 3^{7} a^{2}={ }^{9} C_{3} \times 3^{6} a^{3}$
$\Rightarrow a=\frac{{ }^{9} C_{2}}{{ }^{9} C_{3}} \times 3$
$=\frac{9 ! \times 3 ! \times 6 ! \times 3}{2 ! \times 7 ! \times 9 !}$
$=\frac{9}{7}$