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

Options

A. `3/5`
B. `18/19`
C. `23/38`
D. `27/38`

Answer

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

### Step 1: Determine the Total Number of Guests
There are 5 Indian couples and 5 Russian couples, which means there are a total of:
\[ 10 \text{ couples} \times 2 \text{ (husband and wife)} = 20 \text{ people} \]

### Step 2: Calculate the Total Ways to Shake Hands
The total number of ways to choose 2 people from 20 to shake hands is given by the combination formula:
\[ \binom{20}{2} = \frac{20 \times 19}{2 \times 1} = 190 \]
So, there are 190 total possible handshakes.

### Step 3: Calculate the Handshakes Where a Wife Shakes Hands with Her Husband
Each couple consists of one husband and one wife. If we consider the scenario where at least one wife shakes hands with her husband, we can calculate the number of ways this can happen. Since there are 5 couples, there are 5 ways (one for each couple) for a wife to shake hands with her husband.

### Step 4: Calculate the Handshakes Where No Wife Shakes Hands with Her Husband
To find the number of ways where no wife shakes hands with her husband, we need to subtract the number of ways where at least one wife shakes hands with her husband from the total handshakes:
\[ \text{Ways where no wife shakes hands with her husband} = 190 - 5 = 185 \]

### Step 5: Calculate the Probability
The probability that no wife shakes hands with her husband is given by the ratio of the number of favorable outcomes to the total outcomes:
\[ P(\text{no wife shakes hands with husband}) = \frac{\text{Ways where no wife shakes hands with her husband}}{\text{Total ways to shake hands}} = \frac{185}{190} \]
This simplifies to:
\[ P = \frac{37}{38} \]

### Final Answer
Thus, the probability that no wife shakes hands with her husband is:
\[ \frac{37}{38} \]