Questions: Find the coefficient of (x^3 y^8) in the binomial expansion of ((5 x-2 y)^11). 1650 -1650 5,280,000 -5,280,000

Solution Steps

We are tasked with finding the coefficient of \( x^3 y^8 \) in the binomial expansion of \( (5x - 2y)^{11} \). The general term in the binomial expansion of \( (a + b)^n \) is given by:

\[ T(k) = \binom{n}{k} a^{n-k} b^k \]

where \( a = 5x \), \( b = -2y \), and \( n = 11 \).

To find the coefficient of \( x^3 y^8 \), we need to identify the values of \( k \) and \( n-k \) such that:

\[ n - k = 3 \quad \text{and} \quad k = 8 \]

From \( n = 11 \), we have \( k = 8 \) and \( n - k = 3 \).

Using the values of \( n \) and \( k \), we can calculate the coefficient:

\[ \text{Coefficient} = \binom{11}{8} (5)^{3} (-2)^{8} \]

Calculating each part:

1. \( \binom{11}{8} = \binom{11}{3} = 165 \)
2. \( (5)^{3} = 125 \)
3. \( (-2)^{8} = 256 \)

Now, we can compute the coefficient:

\[ \text{Coefficient} = 165 \times 125 \times 256 \]

Calculating this gives:

\[ \text{Coefficient} = 5280000 \]

Final Answer