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 generating functions.

### Step 1: Define the generating function
The scores possible for each shot can be represented by the generating function:
\[ f(x) = x^0 + x^1 + x^2 + x^3 + x^4 + x^5 \]
This function represents the scores from 0 to 5 points.

### Step 2: Raise the generating function to the power of 7
Since there are 7 shots, we raise the generating function to the power of 7:
\[ f(x)^7 = (x^0 + x^1 + x^2 + x^3 + x^4 + x^5)^7 \]

### Step 3: Simplify the generating function
We can rewrite the generating function as:
\[ f(x) = \frac{1 - x^6}{1 - x} \]
Thus,
\[ f(x)^7 = \left(\frac{1 - x^6}{1 - x}\right)^7 = (1 - x^6)^7 (1 - x)^{-7} \]

### Step 4: Expand using the binomial theorem
Using the binomial theorem, we can expand both parts:
1. \((1 - x^6)^7\) can be expanded as:
\[ \sum_{k=0}^{7} \binom{7}{k} (-1)^k x^{6k} \]
2. \((1 - x)^{-7}\) can be expanded as:
\[ \sum_{m=0}^{\infty} \binom{m + 6}{6} x^m \]

### Step 5: Find the coefficient of \(x^{10}\)
To find the coefficient of \(x^{10}\) in \(f(x)^7\), we need to find the terms where \(6k + m = 10\). This gives us:
\[ m = 10 - 6k \]
The possible values for \(k\) are 0, 1, and 2 (since \(6k\) must be less than or equal to 10).

#### Case 1: \(k = 0\)
\[ m = 10 \implies \text{Coefficient} = \binom{7}{0} \binom{10 + 6}{6} = 1 \cdot \binom{16}{6} = 8008 \]

#### Case 2: \(k = 1\)
\[ m = 4 \implies \text{Coefficient} = \binom{7}{1} (-1)^1 \binom{4 + 6}{6} = -7 \cdot \binom{10}{6} = -7 \cdot 210 = -1470 \]

#### Case 3: \(k = 2\)
\[ m = -2 \implies \text{Coefficient} = \binom{7}{2} (-1)^2 \binom{-2 + 6}{6} = 21 \cdot 0 = 0 \]

### Step 6: Sum the coefficients
Now we sum the coefficients from all cases:
\[ 8008 - 1470 + 0 = 6538 \]

### Final Answer
Thus, the number of different ways in which he can score 10 in 7 shots is:
\[ \boxed{6538} \]