In the expression of `(1+x+x^(3)+x^(4))^(10)`, the coefficient of `x^(4)` is k then number of prime divisors of k is equal to

Explanation

To find the coefficient of \( x^4 \) in the expression \( (1 + x + x^3 + x^4)^{10} \), we can use the multinomial expansion.

### Step-by-step Solution:

1. **Rewrite the Expression**:
We start with the expression:
\[ (1 + x + x^3 + x^4)^{10} \]
We can express this as:
\[ (1 + x)^{10} \cdot (1 + x^3)^{10} \]

2. **Identify Coefficients**:
We need to find the coefficient of \( x^4 \) in the expansion of \( (1 + x)^{10} \) and \( (1 + x^3)^{10} \).

3. **Coefficient of \( x^4 \) in \( (1 + x)^{10} \)**:
The coefficient of \( x^4 \) in \( (1 + x)^{10} \) is given by:
\[ \binom{10}{4} \]

4. **Coefficient of \( x^0 \) in \( (1 + x^3)^{10} \)**:
The coefficient of \( x^0 \) in \( (1 + x^3)^{10} \) is:
\[ \binom{10}{0} = 1 \]

5. **Coefficient of \( x^1 \) in \( (1 + x^3)^{10} \)**:
The coefficient of \( x^1 \) in \( (1 + x^3)^{10} \) is:
\[ \binom{10}{1} = 10 \]

6. **Coefficient of \( x^4 \) in Total**:
Now we can combine these to find the total coefficient of \( x^4 \):
\[ \text{Coefficient of } x^4 = \binom{10}{4} \cdot \binom{10}{0} + \binom{10}{1} \cdot \binom{10}{3} \]

7. **Calculating Each Coefficient**:
- Calculate \( \binom{10}{4} \):
\[ \binom{10}{4} = \frac{10!}{4!(10-4)!} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210 \]
- Calculate \( \binom{10}{3} \):
\[ \binom{10}{3} = \frac{10!}{3!(10-3)!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120 \]

8. **Final Calculation**:
Now substitute back into the equation:
\[ \text{Coefficient of } x^4 = 210 \cdot 1 + 10 \cdot 120 = 210 + 1200 = 1410 \]

9. **Finding Prime Divisors of \( k \)**:
We have \( k = 1410 \). We can factor \( 1410 \):
\[ 1410 = 2 \times 3 \times 5 \times 47 \]
The prime divisors are \( 2, 3, 5, \) and \( 47 \).

10. **Count the Prime Divisors**:
The number of distinct prime divisors of \( k \) is \( 4 \).

### Final Answer:
The number of prime divisors of \( k \) is \( 4 \).