To find the coefficient of the [tex]\( x^6 y^3 \)[/tex] term in the expansion of [tex]\( (x + 2y)^9 \)[/tex], we can use the binomial theorem.
The binomial theorem states:
[tex]\[
(x + a)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} a^k
\][/tex]
For the term [tex]\( x^a y^b \)[/tex] in the expansion:
[tex]\[
(x + 2y)^9 = \sum_{k=0}^{9} \binom{9}{k} x^{9-k} (2y)^k
\][/tex]
We need to identify the coefficient for the term where [tex]\( x \)[/tex] is raised to the 6th power and [tex]\( y \)[/tex] is raised to the 3rd power.
Given:
- The power of [tex]\( x \)[/tex]: [tex]\( x^6 \)[/tex]
- The power of [tex]\( y \)[/tex]: [tex]\( y^3 \)[/tex]
- The total power [tex]\( n \)[/tex]: 9
So, we equate:
[tex]\[
9 - k = 6 \quad \text{and} \quad k = 3
\][/tex]
Now we calculate the binomial coefficient and the term's coefficient:
1. Binomial coefficient [tex]\( \binom{9}{6} \)[/tex]:
[tex]\[
\binom{9}{6} = \binom{9}{3} = \frac{9!}{3!(9-3)!}
= \frac{9 \times 8 \times 7}{3 \times 2 \times 1}
= 84
\][/tex]
2. The term involving [tex]\( (2y)^3 \)[/tex]:
[tex]\[
(2y)^3 = (2^3)(y^3) = 8y^3
\][/tex]
3. The full coefficient of [tex]\( x^6 y^3 \)[/tex]:
[tex]\[
84 \times 8 = 672
\][/tex]
Thus, the coefficient of the [tex]\( x^6 y^3 \)[/tex] term in the expansion of [tex]\( (x + 2y)^9 \)[/tex] is:
[tex]\[
\boxed{672}
\][/tex]
