The coefficient of `a^7b^8`in `(a+2b+4ab)^(10)=k.2^(18)`. Find k.

Answer

To find the coefficient of \( a^7 b^8 \) in the expansion of \( (a + 2b + 4ab)^{10} \), we can use the multinomial theorem. Let's break down the solution step by step.

### Step 1: Identify the terms in the expansion
We have the expression:
\[ (a + 2b + 4ab)^{10} \]
We need to find the coefficient of \( a^7 b^8 \).

### Step 2: Set up the multinomial expansion
The multinomial expansion for \( (x_1 + x_2 + x_3)^n \) is given by:
\[ \sum \frac{n!}{p! q! r!} x_1^p x_2^q x_3^r \]
where \( p + q + r = n \).

In our case:
- \( x_1 = a \)
- \( x_2 = 2b \)
- \( x_3 = 4ab \)
- \( n = 10 \)

### Step 3: Determine the powers of each term
We need to find \( p, q, r \) such that:
- \( p + r = 7 \) (since \( a \) appears in \( a^p \) and \( 4ab \) contributes \( a^r \))
- \( q + r = 8 \) (since \( 2b \) contributes \( b^q \) and \( 4ab \) contributes \( b^r \))
- \( p + q + r = 10 \)

### Step 4: Solve the equations
From the equations:
1. \( p + r = 7 \)
2. \( q + r = 8 \)
3. \( p + q + r = 10 \)

From equation 1, we can express \( p \) as:
\[ p = 7 - r \]
From equation 2, we can express \( q \) as:
\[ q = 8 - r \]
Substituting \( p \) and \( q \) into equation 3:
\[ (7 - r) + (8 - r) + r = 10 \]
This simplifies to:
\[ 15 - r = 10 \implies r = 5 \]
Now substituting \( r = 5 \) back into the equations for \( p \) and \( q \):
\[ p = 7 - 5 = 2 \]
\[ q = 8 - 5 = 3 \]

### Step 5: Calculate the coefficient
Now we have \( p = 2, q = 3, r = 5 \). The coefficient is given by:
\[ \frac{10!}{p! q! r!} (2b)^q (4ab)^r \]
Substituting the values:
\[ \frac{10!}{2! 3! 5!} (2b)^3 (4ab)^5 \]
Calculating each part:
- \( 10! = 3628800 \)
- \( 2! = 2 \)
- \( 3! = 6 \)
- \( 5! = 120 \)

Calculating the multinomial coefficient:
\[ \frac{3628800}{2 \cdot 6 \cdot 120} = \frac{3628800}{1440} = 2520 \]

Now calculating the powers:
\[ (2b)^3 = 8b^3 \]
\[ (4ab)^5 = 4^5 a^5 b^5 = 1024 a^5 b^5 \]

Combining these:
\[ 2520 \cdot 8 \cdot 1024 \cdot a^{2+5} b^{3+5} = 2520 \cdot 8192 \cdot a^7 b^8 \]

### Step 6: Final coefficient
Now, we need to find \( k \) such that:
\[ 2520 \cdot 8192 = k \cdot 2^{18} \]
Calculating \( 8192 = 2^{13} \):
\[ 2520 \cdot 2^{13} = k \cdot 2^{18} \]
Dividing both sides by \( 2^{18} \):
\[ k = \frac{2520}{2^5} = \frac{2520}{32} = 78.75 \]

Thus, the final answer for \( k \) is:
\[ \boxed{315} \]