If ` (x + a_(1)) (x + a_(2)) (x + a_(3)) …(x + a_(n)) = x^(n) + S_(1) x^(n-1) + S_(2) x^(n-2) + …+ S_(n)`
where ,` S_(1) = sum_(i=0)^(n) a_(i), S_(2) = (sumsum)_(1lei lt j le n) a_(i) a_(j) , S_(3) (sumsumsum)_(1le i ltk le n)a_(i) a_(j) a_(k)`
and so on .
Coefficient of ` x^(7)` in the expansion of
` (1 + x)^(2) (3 + x)^(3) (5 + x)^(4)` is

Options

A. 112
B. 224
C. 342
D. 416

Explanation

To find the coefficient of \( x^7 \) in the expansion of \( (1 + x)^2 (3 + x)^3 (5 + x)^4 \), we can follow these steps:

### Step 1: Expand Each Factor
We start by expanding each factor separately using the binomial theorem.

1. **For \( (1 + x)^2 \)**:
\[ (1 + x)^2 = 1^2 + \binom{2}{1} x + \binom{2}{2} x^2 = 1 + 2x + x^2 \]

2. **For \( (3 + x)^3 \)**:
\[ (3 + x)^3 = \sum_{k=0}^{3} \binom{3}{k} 3^{3-k} x^k = 27 + 27x + 9x^2 + x^3 \]

3. **For \( (5 + x)^4 \)**:
\[ (5 + x)^4 = \sum_{k=0}^{4} \binom{4}{k} 5^{4-k} x^k = 625 + 500x + 150x^2 + 20x^3 + x^4 \]

### Step 2: Combine the Expansions
Now, we need to find the coefficient of \( x^7 \) in the product of these three expansions:
\[ (1 + 2x + x^2)(27 + 27x + 9x^2 + x^3)(625 + 500x + 150x^2 + 20x^3 + x^4) \]

### Step 3: Identify Terms Contributing to \( x^7 \)
To find the coefficient of \( x^7 \), we consider combinations of terms from each factor that add up to \( x^7 \):

1. **From \( (1 + 2x + x^2) \)**:
- Take \( 1 \) from \( (1 + 2x + x^2) \) and find terms from the other two factors that sum to \( x^7 \).
- Take \( 2x \) from \( (1 + 2x + x^2) \) and find terms from the other two factors that sum to \( x^6 \).
- Take \( x^2 \) from \( (1 + 2x + x^2) \) and find terms from the other two factors that sum to \( x^5 \).

### Step 4: Calculate Each Contribution
1. **Contribution from \( 1 \)**:
- From \( (27 + 27x + 9x^2 + x^3) \) and \( (625 + 500x + 150x^2 + 20x^3 + x^4) \):
- \( x^7 \) term: \( 0 \) (no \( x^7 \) term)

2. **Contribution from \( 2x \)**:
- From \( (27 + 27x + 9x^2 + x^3) \) and \( (625 + 500x + 150x^2 + 20x^3 + x^4) \):
- \( x^6 \) term: \( 0 \) (no \( x^6 \) term)

3. **Contribution from \( x^2 \)**:
- From \( (27 + 27x + 9x^2 + x^3) \) and \( (625 + 500x + 150x^2 + 20x^3 + x^4) \):
- \( x^5 \) term: \( 20 \) (from \( 20x^3 \))
- Coefficient: \( 1 \cdot 20 = 20 \)

### Step 5: Final Calculation
Now we sum up all contributions:
- From \( 1 \): \( 0 \)
- From \( 2x \): \( 0 \)
- From \( x^2 \): \( 20 \)

Thus, the coefficient of \( x^7 \) is:
\[ \text{Coefficient of } x^7 = 20 \]

### Final Answer
The coefficient of \( x^7 \) in the expansion of \( (1 + x)^2 (3 + x)^3 (5 + x)^4 \) is \( 20 \).

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