Binomial Theorem: Exam-Pattern Drill (4)

For $(a + b)^n$, the general term is:

$T_{r+1} = \binom{n}{r} a^{n-r} b^r$

For our expansion with $a = 2x$, $b = 3/x^2$, $n = 15$:

$T_{r+1} = \binom{15}{r} (2x)^{15-r} \left(\frac{3}{x^2}\right)^r = \binom{15}{r} 2^{15-r} \cdot 3^r \cdot x^{15-r-2r} = \binom{15}{r} 2^{15-r} 3^r \cdot x^{15-3r}$

For the constant term, the power of $x$ must be zero:

$15 - 3r = 0 \implies r = 5$

Substituting $r = 5$:

$T_6 = \binom{15}{5} 2^{10} \cdot 3^5 = 3003 \cdot 1024 \cdot 243 = 3003 \cdot 248832$

Computing: $\binom{15}{5} = 3003$. So the constant term is $3003 \cdot 1024 \cdot 243$, a specific number — leave in product form for cleanliness.

Set $15 - 3r = 9 \implies r = 2$.

$T_3 = \binom{15}{2} 2^{13} \cdot 3^2 \cdot x^9 = 105 \cdot 8192 \cdot 9 \cdot x^9 = 7741440 \cdot x^9$

So the coefficient of $x^9$ is $\binom{15}{2} \cdot 2^{13} \cdot 9 = 7,741,440$.

(a) general term as above, (b) constant term at $r = 5$, (c) coefficient of $x^9 = 7,741,440$.