In how many ways 10 persons can be divided into 5 pairs?

Answer

To find the number of ways to divide 10 persons into 5 pairs, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**: We need to divide 10 persons into 5 pairs. Each pair consists of 2 persons.

2. **Choosing the First Pair**: We can choose the first pair from 10 persons. The number of ways to choose 2 persons from 10 is given by the combination formula:
\[ \binom{10}{2} = \frac{10!}{2!(10-2)!} = \frac{10 \times 9}{2 \times 1} = 45 \]

3. **Choosing the Second Pair**: After choosing the first pair, we have 8 persons left. The number of ways to choose the second pair from these 8 persons is:
\[ \binom{8}{2} = \frac{8!}{2!(8-2)!} = \frac{8 \times 7}{2 \times 1} = 28 \]

4. **Choosing the Third Pair**: Now, we have 6 persons left. The number of ways to choose the third pair is:
\[ \binom{6}{2} = \frac{6!}{2!(6-2)!} = \frac{6 \times 5}{2 \times 1} = 15 \]

5. **Choosing the Fourth Pair**: We now have 4 persons left. The number of ways to choose the fourth pair is:
\[ \binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6 \]

6. **Choosing the Fifth Pair**: Finally, we have 2 persons left, which will automatically form the last pair:
\[ \binom{2}{2} = 1 \]

7. **Calculating the Total Combinations**: Now, we multiply the number of ways to choose each pair:
\[ \text{Total ways} = \binom{10}{2} \times \binom{8}{2} \times \binom{6}{2} \times \binom{4}{2} \times \binom{2}{2} = 45 \times 28 \times 15 \times 6 \times 1 \]

8. **Adjusting for Overcounting**: Since the order of the pairs does not matter, we have counted each arrangement of pairs multiple times. We need to divide by the number of ways to arrange 5 pairs, which is \(5!\):
\[ 5! = 120 \]

9. **Final Calculation**: Therefore, the total number of ways to divide 10 persons into 5 pairs is:
\[ \text{Total ways} = \frac{45 \times 28 \times 15 \times 6 \times 1}{120} \]

10. **Calculating the Result**:
\[ 45 \times 28 = 1260 \]
\[ 1260 \times 15 = 18900 \]
\[ 18900 \times 6 = 113400 \]
\[ \frac{113400}{120} = 945 \]

Thus, the total number of ways to divide 10 persons into 5 pairs is **945**.