Find the coefficient of `x^(2).y^(7)` in the expansion of `(x+2y)^(9)`

Explanation

To find the coefficient of \( x^2 y^7 \) in the expansion of \( (x + 2y)^9 \), we can use the Binomial Theorem. The Binomial Theorem states that:

\[ (a + b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^r \]

In our case, we have \( a = x \), \( b = 2y \), and \( n = 9 \). The general term in the expansion can be expressed as:

\[ T_{r+1} = \binom{9}{r} x^{9-r} (2y)^r \]

This simplifies to:

\[ T_{r+1} = \binom{9}{r} x^{9-r} 2^r y^r \]

We are interested in the term where \( x^{9-r} = x^2 \) and \( y^r = y^7 \). This gives us two equations:

1. \( 9 - r = 2 \)
2. \( r = 7 \)

From the first equation, we can solve for \( r \):

\[ 9 - r = 2 \implies r = 7 \]

Now, we can substitute \( r = 7 \) back into the general term to find the coefficient:

\[ T_{8} = \binom{9}{7} x^{2} (2y)^7 \]

Calculating \( \binom{9}{7} \):

\[ \binom{9}{7} = \binom{9}{2} = \frac{9 \times 8}{2 \times 1} = 36 \]

Now, substituting \( r = 7 \) into \( (2y)^7 \):

\[ (2y)^7 = 2^7 y^7 = 128 y^7 \]

Now we can find the coefficient of \( x^2 y^7 \):

\[ \text{Coefficient} = \binom{9}{7} \cdot 2^7 = 36 \cdot 128 \]

Calculating \( 36 \cdot 128 \):

\[ 36 \cdot 128 = 4608 \]

Thus, the coefficient of \( x^2 y^7 \) in the expansion of \( (x + 2y)^9 \) is:

\[ \boxed{4608} \]