The coefficient of `x^50` in `(1+x^2)^25(1+x^25)(1+x^40)(1+ x^45) (1 + x^47)`, is

Options

A. `""^(25)C_(5) +1`
B. `""^(25)C_(5) + ""^(25)C_(7+1)`
C. `""^(25)C_(7) +1`
D. none of these

Explanation

To find the coefficient of \( x^{50} \) in the expression

\[ (1 + x^2)^{25} (1 + x^{25})(1 + x^{40})(1 + x^{45})(1 + x^{47}), \]

we can break down the problem step by step.

### Step 1: Expand \( (1 + x^2)^{25} \)

The binomial expansion of \( (1 + x^2)^{25} \) is given by:

\[ (1 + x^2)^{25} = \sum_{k=0}^{25} \binom{25}{k} (x^2)^k = \sum_{k=0}^{25} \binom{25}{k} x^{2k}. \]

### Step 2: Identify terms contributing to \( x^{50} \)

We need to find terms from the expansion of \( (1 + x^2)^{25} \) that can combine with other terms from \( (1 + x^{25})(1 + x^{40})(1 + x^{45})(1 + x^{47}) \) to give \( x^{50} \).

1. **From \( (1 + x^2)^{25} \)**:
- The term \( x^{50} \) can be obtained when \( k = 25 \) (i.e., \( 2k = 50 \)). The coefficient for this term is \( \binom{25}{25} = 1 \).

2. **From \( (1 + x^{25})(1 + x^{40})(1 + x^{45})(1 + x^{47}) \)**:
- We need to find combinations of terms that add up to \( 50 \).

### Step 3: Find combinations of terms

We can choose terms from \( (1 + x^{25})(1 + x^{40})(1 + x^{45})(1 + x^{47}) \):

- **Choose \( x^{40} \)** and \( x^{10} \) (which we can get from \( (1 + x^2)^{25} \)):
- We need \( 2k = 10 \) which gives \( k = 5 \). The coefficient is \( \binom{25}{5} \).

- **Choose \( x^{25} \)** and \( x^{25} \) (from \( (1 + x^{25})^2 \)):
- We need \( 2k = 50 \) which gives \( k = 25 \). The coefficient is \( \binom{25}{25} = 1 \).

- **Choose \( x^{45} \)** and \( x^{5} \) (which we can get from \( (1 + x^2)^{25} \)):
- We need \( 2k = 5 \) which gives \( k = 2.5 \) (not possible).

- **Choose \( x^{47} \)** and \( x^{3} \) (which we can get from \( (1 + x^2)^{25} \)):
- We need \( 2k = 3 \) which gives \( k = 1.5 \) (not possible).

### Step 4: Calculate the total coefficient

The valid contributions to \( x^{50} \) are:

1. From \( k = 25 \) (coefficient = 1).
2. From \( k = 5 \) (coefficient = \( \binom{25}{5} \)).

Thus, the total coefficient of \( x^{50} \) is:

\[ 1 + \binom{25}{5}. \]

### Step 5: Calculate \( \binom{25}{5} \)

Calculating \( \binom{25}{5} \):

\[ \binom{25}{5} = \frac{25 \times 24 \times 23 \times 22 \times 21}{5 \times 4 \times 3 \times 2 \times 1} = 53130. \]

### Final Answer

Thus, the coefficient of \( x^{50} \) is:

\[ 1 + 53130 = 53131. \]