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, then the probability that he gets two free packets is

Options

A. `1//7`
B. `1//5`
C. `1//42`
D. `1//165`

Explanation

To solve the problem, we need to find the probability that a person who buys 8 packets of detergent SURF will get 2 free packets. To do this, we will follow these steps:

### Step 1: Understanding the Problem
Each packet of detergent SURF contains a coupon with one of the letters from the word "SURF". To get a free packet, a person must collect all four letters (S, U, R, F) at least once. If he buys 8 packets, we need to determine how many ways he can collect the letters to get 2 free packets.

### Step 2: Total Ways to Distribute Letters
We need to find the total number of ways to distribute the letters among the 8 packets. Let \( x_1, x_2, x_3, x_4 \) represent the number of times the letters S, U, R, and F appear respectively. The equation we need to solve is:

\[ x_1 + x_2 + x_3 + x_4 = 8 \]

where \( x_1, x_2, x_3, x_4 \geq 0 \).

Using the "stars and bars" theorem, the number of non-negative integral solutions to this equation is given by:

\[ \binom{n + r - 1}{r - 1} \]

where \( n = 8 \) (the total number of packets) and \( r = 4 \) (the number of different letters). Thus, we have:

\[ \text{Total ways} = \binom{8 + 4 - 1}{4 - 1} = \binom{11}{3} = 165 \]

### Step 3: Ways to Get 2 Free Packets
To get 2 free packets, the person must collect all four letters at least twice. This means we need to distribute the letters such that each letter appears at least 2 times. Let’s denote the new variables:

Let \( y_1 = x_1 - 2 \), \( y_2 = x_2 - 2 \), \( y_3 = x_3 - 2 \), \( y_4 = x_4 - 2 \).

Now, the equation becomes:

\[ y_1 + y_2 + y_3 + y_4 = 8 - 8 = 0 \]

The only solution to this equation is \( y_1 = y_2 = y_3 = y_4 = 0 \). Thus, the only way to have exactly 2 of each letter (S, U, R, F) is:

\[ x_1 = 2, x_2 = 2, x_3 = 2, x_4 = 2 \]

### Step 4: Counting the Ways to Achieve This
The number of ways to arrange the letters S, U, R, F, each appearing exactly 2 times in 8 positions is given by:

\[ \frac{8!}{2! \times 2! \times 2! \times 2!} = \frac{40320}{16} = 2520 \]

### Step 5: Calculating the Probability
Now, we can calculate the probability of getting 2 free packets:

\[ \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{2520}{165} \]

### Step 6: Simplifying the Probability
To simplify:

\[ \frac{2520}{165} = \frac{504}{33} = \frac{168}{11} \approx 15.27 \]

However, since we are looking for a probability, we need to express it in a fraction form.

### Final Answer
The probability that he gets 2 free packets is:

\[ \frac{2520}{165} \text{ or approximately } 15.27 \]