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

Answer

To solve the problem of dividing 10 persons into 5 pairs, we can follow these steps:

### Step 1: Understand the Problem
We need to pair up 10 individuals into 5 distinct pairs. Each pair consists of 2 persons.

### Step 2: Calculate Total Arrangements
First, we find the total number of ways to arrange 10 persons. This can be calculated using the factorial of 10, denoted as \(10!\).

\[ 10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \]

### Step 3: Account for Pairing
Since we are forming pairs, each pair can be arranged in \(2!\) ways (i.e., the order of the two persons in a pair does not matter). Since there are 5 pairs, we need to divide by \(2^5\) (which is \(2!\) for each of the 5 pairs).

\[ 2^5 = 32 \]

### Step 4: Account for Indistinguishable Pairs
The pairs themselves are indistinguishable from one another. Therefore, we also need to divide by the number of ways to arrange the 5 pairs, which is \(5!\).

\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]

### Step 5: Combine the Results
Now we can combine all these calculations into one formula:

\[ \text{Number of ways} = \frac{10!}{2^5 \times 5!} \]

### Step 6: Calculate the Values
Now we substitute the values we calculated:

1. Calculate \(10!\):
\[ 10! = 3628800 \]

2. Calculate \(2^5\):
\[ 2^5 = 32 \]

3. Calculate \(5!\):
\[ 5! = 120 \]

Now substitute these values into the formula:

\[ \text{Number of ways} = \frac{3628800}{32 \times 120} \]

### Step 7: Simplify the Denominator
Calculate the denominator:

\[ 32 \times 120 = 3840 \]

### Step 8: Final Calculation
Now perform the division:

\[ \text{Number of ways} = \frac{3628800}{3840} = 945 \]

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

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