5 Indian and 5 Russian couples meet at a party and shake hands.
The probability that no Indian wife shakes hands with a male is

Options

A. `14/19`
B. `9/19`
C. `27/38`
D. `27/28`

Answer

To solve the problem of finding the probability that no Indian wife shakes hands with a male at a party where there are 5 Indian and 5 Russian couples, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Total Number of People:**
- There are 5 Indian couples and 5 Russian couples, which means there are:
\[ 5 \text{ Indian husbands} + 5 \text{ Indian wives} + 5 \text{ Russian husbands} + 5 \text{ Russian wives} = 20 \text{ people} \]

2. **Calculate the Total Number of Handshakes:**
- The total number of ways to choose 2 people from 20 to shake hands is given by the combination formula \( \binom{n}{r} \):
\[ \text{Total handshakes} = \binom{20}{2} = \frac{20 \times 19}{2} = 190 \]

3. **Determine the Handshakes Involving Indian Wives:**
- Each Indian wife can shake hands with any of the 10 males (5 Indian husbands + 5 Russian husbands). Therefore, the number of handshakes involving Indian wives is:
\[ \text{Handshakes by Indian wives} = 5 \text{ Indian wives} \times 10 \text{ males} = 50 \]

4. **Calculate the Number of Handshakes Where No Indian Wife Shakes Hands with a Male:**
- If we want to find the scenarios where no Indian wife shakes hands with any male, we subtract the handshakes involving Indian wives from the total handshakes:
\[ \text{Handshakes where no Indian wife shakes hands with males} = 190 - 50 = 140 \]

5. **Calculate the Probability:**
- The probability that no Indian wife shakes hands with a male is the number of favorable outcomes divided by the total outcomes:
\[ P(\text{no Indian wife shakes hands with a male}) = \frac{140}{190} = \frac{14}{19} \]

Thus, the probability that no Indian wife shakes hands with a male is \( \frac{14}{19} \).