A shoping mall is running a scheme: Each packet of detergent SURF contains a coupon which bears letter of the word SURF, if a person buys at least four packets of detergent SURF, and produce all the letters of the word SURF, then he gets one free packet of detergent.
If a person buys 8 such packets at a time, then the number of different combinations of coupon he has is

Options

A. `4^(8)`
B. `8^(4)`
C. `.^(11)C_(3)`
D. `.^(12)C_(4)`

Explanation

To solve the problem, we need to find the number of different combinations of coupons that a person can have when buying 8 packets of detergent SURF, each containing a coupon with one of the letters S, U, R, F.

### Step-by-Step Solution:

1. **Understanding the Problem**:
Each packet of detergent SURF contains one of the letters: S, U, R, or F. If a person buys 8 packets, we need to determine how many different combinations of these letters (coupons) can be formed.

2. **Setting Up the Equation**:
Let \( x_1 \) be the number of S coupons, \( x_2 \) be the number of U coupons, \( x_3 \) be the number of R coupons, and \( x_4 \) be the number of F coupons. The total number of packets (coupons) bought is 8, which gives us the equation:
\[ x_1 + x_2 + x_3 + x_4 = 8 \]
where \( x_1, x_2, x_3, x_4 \geq 0 \).

3. **Using the Stars and Bars Theorem**:
The problem can be solved using the "stars and bars" theorem, which is a popular combinatorial method for distributing indistinguishable objects (in this case, the coupons) into distinguishable boxes (the letters S, U, R, F).

4. **Applying the Formula**:
According to the stars and bars theorem, the number of non-negative integer solutions to the equation \( x_1 + x_2 + x_3 + x_4 = n \) is given by:
\[ \binom{n + r - 1}{r - 1} \]
where \( n \) is the total number of items (coupons) and \( r \) is the number of categories (letters). Here, \( n = 8 \) and \( r = 4 \).

5. **Calculating the Combinations**:
Plugging in the values:
\[ \text{Number of combinations} = \binom{8 + 4 - 1}{4 - 1} = \binom{11}{3} \]

6. **Calculating \( \binom{11}{3} \)**:
To calculate \( \binom{11}{3} \):
\[ \binom{11}{3} = \frac{11 \times 10 \times 9}{3 \times 2 \times 1} = \frac{990}{6} = 165 \]

7. **Final Answer**:
Therefore, the number of different combinations of coupons when a person buys 8 packets of detergent SURF is **165**.