Mr. Daruwala has 18 acquaintances of whom 13 are relatives. In how many ways he may invite 10 guests so that 8 of whom are relatives ?

Options

A. 12870
B. 22022
C. 20222
D. 12780

Answer

To solve the problem of how many ways Mr. Daruwala can invite 10 guests with 8 being relatives, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Total Number of Acquaintances**:
Mr. Daruwala has a total of 18 acquaintances, which includes 13 relatives and 5 non-relatives.

2. **Determine the Number of Guests to Invite**:
He wants to invite a total of 10 guests, out of which 8 must be relatives.

3. **Calculate the Number of Non-Relatives to Invite**:
Since 8 guests are relatives, the remaining guests must be non-relatives. Therefore, he will invite:
\[ 10 - 8 = 2 \text{ non-relatives} \]

4. **Calculate the Combinations for Relatives**:
To find out how many ways he can choose 8 relatives from 13, we use the combination formula:
\[ \text{Number of ways to choose 8 relatives from 13} = \binom{13}{8} \]
Using the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
We can calculate:
\[ \binom{13}{8} = \frac{13!}{8! \cdot (13-8)!} = \frac{13!}{8! \cdot 5!} \]

5. **Calculate the Combinations for Non-Relatives**:
Now, we need to choose 2 non-relatives from the 5 available:
\[ \text{Number of ways to choose 2 non-relatives from 5} = \binom{5}{2} \]
Again using the combination formula:
\[ \binom{5}{2} = \frac{5!}{2! \cdot (5-2)!} = \frac{5!}{2! \cdot 3!} \]

6. **Calculate Each Combination**:
- For relatives:
\[ \binom{13}{8} = \frac{13 \times 12 \times 11 \times 10}{4 \times 3 \times 2 \times 1} = 1287 \]
- For non-relatives:
\[ \binom{5}{2} = \frac{5 \times 4}{2 \times 1} = 10 \]

7. **Multiply the Two Results**:
The total number of ways to invite the guests is the product of the two combinations:
\[ \text{Total ways} = \binom{13}{8} \times \binom{5}{2} = 1287 \times 10 = 12870 \]

### Final Answer:
Thus, the total number of ways Mr. Daruwala can invite 10 guests, with 8 being relatives, is **12870**.

---