Find the coefficient of `a^5b^7in(a-2b)^(12)`

Explanation

To find the coefficient of \( a^5b^7 \) in the expansion of \( (a - 2b)^{12} \), we can follow these steps:

### Step 1: Identify the General Term
The general term in the binomial expansion of \( (p + q)^n \) is given by:
\[ T_r = \binom{n}{r} p^{n-r} q^r \]
For our case, we have \( p = a \), \( q = -2b \), and \( n = 12 \). Therefore, the general term becomes:
\[ T_r = \binom{12}{r} a^{12-r} (-2b)^r \]

### Step 2: Simplify the General Term
Now, we simplify the general term:
\[ T_r = \binom{12}{r} a^{12-r} (-2)^r b^r \]
This can be rewritten as:
\[ T_r = \binom{12}{r} (-2)^r a^{12-r} b^r \]

### Step 3: Set Up the Conditions for \( a^5b^7 \)
We need to find the term where the power of \( a \) is 5 and the power of \( b \) is 7. This gives us the following equations:
1. \( 12 - r = 5 \) (for \( a \))
2. \( r = 7 \) (for \( b \))

### Step 4: Solve for \( r \)
From the first equation:
\[ 12 - r = 5 \implies r = 7 \]
This confirms our second equation.

### Step 5: Substitute \( r \) into the General Term
Now, we substitute \( r = 7 \) into the general term:
\[ T_7 = \binom{12}{7} (-2)^7 a^{12-7} b^7 \]
This simplifies to:
\[ T_7 = \binom{12}{7} (-2)^7 a^5 b^7 \]

### Step 6: Calculate the Coefficient
Now, we need to calculate the coefficient:
\[ \text{Coefficient} = \binom{12}{7} (-2)^7 \]
Calculating \( \binom{12}{7} \):
\[ \binom{12}{7} = \binom{12}{5} = \frac{12 \times 11 \times 10 \times 9 \times 8}{5 \times 4 \times 3 \times 2 \times 1} = 792 \]
Now, calculate \( (-2)^7 \):
\[ (-2)^7 = -128 \]
Thus, the coefficient is:
\[ \text{Coefficient} = 792 \times (-128) = -101376 \]

### Final Answer
The coefficient of \( a^5b^7 \) in the expansion of \( (a - 2b)^{12} \) is \( -101376 \).

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