If the coefficient of `x^(6)` in the expansion of `(2+x)^(3)(3+x)^(2)(5+x)^(3)` is K, then the value of `(K)/(100)` is

Answer

To find the coefficient of \( x^6 \) in the expansion of \( (2+x)^3(3+x)^2(5+x)^3 \), we can use the multinomial expansion and the binomial theorem. Let's break down the solution step by step.

### Step 1: Identify the terms contributing to \( x^6 \)

We need to find the coefficient of \( x^6 \) in the expansion. The expression can be expanded as follows:

\[ (2+x)^3 = \sum_{k=0}^{3} \binom{3}{k} 2^{3-k} x^k \]
\[ (3+x)^2 = \sum_{j=0}^{2} \binom{2}{j} 3^{2-j} x^j \]
\[ (5+x)^3 = \sum_{m=0}^{3} \binom{3}{m} 5^{3-m} x^m \]

### Step 2: Determine combinations of \( k, j, m \)

We need to find combinations of \( k, j, m \) such that:

\[ k + j + m = 6 \]

Where \( k \) can take values from 0 to 3, \( j \) can take values from 0 to 2, and \( m \) can take values from 0 to 3.

### Step 3: Calculate contributions for each valid combination

We will consider all combinations of \( k, j, m \) that satisfy \( k + j + m = 6 \):

1. **\( k = 3, j = 2, m = 1 \)**:
- Contribution:
\[ \binom{3}{3} 2^0 \cdot \binom{2}{2} 3^0 \cdot \binom{3}{1} 5^2 = 1 \cdot 1 \cdot 3 \cdot 25 = 75 \]

2. **\( k = 2, j = 2, m = 2 \)**:
- Contribution:
\[ \binom{3}{2} 2^1 \cdot \binom{2}{2} 3^0 \cdot \binom{3}{2} 5^1 = 3 \cdot 2 \cdot 1 \cdot 3 \cdot 5 = 90 \]

3. **\( k = 1, j = 2, m = 3 \)**:
- Contribution:
\[ \binom{3}{1} 2^2 \cdot \binom{2}{2} 3^0 \cdot \binom{3}{3} 5^0 = 3 \cdot 4 \cdot 1 \cdot 1 = 12 \]

4. **\( k = 0, j = 2, m = 4 \)**:
- Contribution:
\[ \binom{3}{0} 2^3 \cdot \binom{2}{2} 3^0 \cdot \binom{3}{4} 5^0 = 1 \cdot 8 \cdot 1 \cdot 1 = 8 \]

### Step 4: Sum the contributions

Now, we sum all contributions:

\[ 75 + 90 + 12 + 8 = 185 \]

### Step 5: Calculate \( K \) and \( \frac{K}{100} \)

Thus, \( K = 185 \). Therefore,

\[ \frac{K}{100} = \frac{185}{100} = 1.85 \]

### Final Answer

The value of \( \frac{K}{100} \) is \( 1.85 \).

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