A man has 12 friends of whom 8 are relatives.How many ways he can invite 7 friends such that 5 of them are relatives is

Options

A. a. 336
B. b. 330
C. c. 202
D. d. 127

Answer

To solve the problem of how many ways a man can invite 7 friends such that 5 of them are relatives, we can follow these steps:

### Step 1: Identify the total number of friends and relatives
- The man has a total of 12 friends.
- Out of these, 8 are relatives and the remaining 4 are non-relatives (friends).

### Step 2: Determine the selection criteria
- We need to invite a total of 7 friends.
- Out of these 7 friends, 5 must be relatives.

### Step 3: Calculate the number of ways to choose the relatives
- We need to select 5 relatives from the 8 available relatives.
- The number of ways to choose 5 relatives from 8 is given by the combination formula \( \binom{n}{r} \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose.

\[ \text{Number of ways to choose 5 relatives} = \binom{8}{5} \]

Using the combination formula:

\[ \binom{8}{5} = \frac{8!}{5!(8-5)!} = \frac{8!}{5!3!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56 \]

### Step 4: Calculate the number of ways to choose the non-relatives
- After selecting 5 relatives, we need to select 2 more friends from the remaining 4 non-relatives.
- The number of ways to choose 2 non-relatives from 4 is:

\[ \text{Number of ways to choose 2 non-relatives} = \binom{4}{2} \]

Using the combination formula:

\[ \binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4!}{2!2!} = \frac{4 \times 3}{2 \times 1} = 6 \]

### Step 5: Calculate the total number of ways to invite the friends
- To find the total number of ways to invite 7 friends (5 relatives and 2 non-relatives), we multiply the number of ways to choose the relatives by the number of ways to choose the non-relatives:

\[ \text{Total ways} = \binom{8}{5} \times \binom{4}{2} = 56 \times 6 = 336 \]

### Final Answer
Thus, the total number of ways the man can invite 7 friends such that 5 of them are relatives is **336**.

---