In a shooting competition a man can score 5, 4, 3, 2, 1 or 0 points for each shot. Then the number of different ways in which he can score 10 in seven shots is

Options

A. 6538
B. 6648
C. 6468
D. 6236

Answer

To solve the problem of finding the number of different ways a man can score 10 points in 7 shots, where he can score 5, 4, 3, 2, 1, or 0 points for each shot, we can use the concept of generating functions or combinatorial counting.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We need to find the number of combinations of scores (5, 4, 3, 2, 1, 0) that sum up to 10 points in 7 shots.

2. **Setting Up the Equation**:
Let \( x_1, x_2, x_3, x_4, x_5, x_6, x_7 \) be the scores from each of the 7 shots. We need to find the number of non-negative integer solutions to the equation:
\[ x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 = 10 \]
where \( x_i \) can take values from the set {0, 1, 2, 3, 4, 5}.

3. **Generating Function**:
The generating function for each shot can be represented as:
\[ f(x) = x^0 + x^1 + x^2 + x^3 + x^4 + x^5 = \frac{1 - x^6}{1 - x} \]
This represents the scores from 0 to 5.

4. **Total Generating Function**:
Since there are 7 shots, the total generating function becomes:
\[ (f(x))^7 = \left(\frac{1 - x^6}{1 - x}\right)^7 \]

5. **Finding Coefficient of \( x^{10} \)**:
We need to find the coefficient of \( x^{10} \) in the expansion of:
\[ (1 - x^6)^7 (1 - x)^{-7} \]

6. **Using Binomial Theorem**:
We can expand \( (1 - x^6)^7 \) using the binomial theorem:
\[ (1 - x^6)^7 = \sum_{k=0}^{7} \binom{7}{k} (-1)^k x^{6k} \]
and \( (1 - x)^{-7} \) can be expanded as:
\[ (1 - x)^{-7} = \sum_{m=0}^{\infty} \binom{m + 6}{6} x^m \]

7. **Combining the Two Expansions**:
We want to find the coefficient of \( x^{10} \) in the product of these two series. This can be done by considering all combinations of \( k \) and \( m \) such that:
\[ 6k + m = 10 \quad \text{or} \quad m = 10 - 6k \]
where \( k \) can take values from 0 to 7.

8. **Calculating Coefficients**:
We need to calculate:
\[ \sum_{k=0}^{\lfloor 10/6 \rfloor} \binom{7}{k} (-1)^k \binom{10 - 6k + 6}{6} \]
- For \( k = 0 \): \( \binom{7}{0} \binom{16}{6} \)
- For \( k = 1 \): \( -\binom{7}{1} \binom{10}{6} \)
- For \( k = 2 \): \( \binom{7}{2} \binom{4}{6} = 0 \) (since \( \binom{4}{6} = 0 \))

9. **Final Calculation**:
- Calculate \( \binom{16}{6} = 8008 \)
- Calculate \( \binom{10}{6} = 210 \)
- Thus, the total becomes:
\[ 8008 - 7 \times 210 = 8008 - 1470 = 6538 \]

### Final Answer:
The number of different ways in which he can score 10 in 7 shots is **6538**.