The total number of ways in which `2n` persons can be divided into `n` couples is a. `(2n !)/(n ! n !)` b. `(2n !)/((2!)^3)` c. `(2n !)/(n !(2!)^n)` d. none of these

Options

A. `(2n!)/((n!)^(2))`
B. `(2n!)/((2n!)^(n))`
C. `(2n!)/(n!(2n!)^(2))`
D. None of these

Answer

To solve the problem of dividing `2n` persons into `n` couples, we can follow these steps:

### Step 1: Understand the problem
We need to divide `2n` persons into `n` couples. Each couple consists of 2 persons.

### Step 2: Calculate the total arrangements of `2n` persons
The total number of ways to arrange `2n` persons is given by `2n!`.

### Step 3: Account for the indistinguishability of couples
Since each couple consists of 2 persons, the arrangement of each couple does not matter. Therefore, for each couple, we can arrange the 2 persons in `2!` ways. Since we have `n` couples, we need to divide by `(2!)^n` to account for the indistinguishable arrangements within each couple.

### Step 4: Account for the indistinguishability of the couples
The `n` couples themselves can be arranged among themselves in `n!` ways. Since the order of the couples does not matter, we also need to divide by `n!`.

### Step 5: Combine the calculations
Putting it all together, the total number of ways to divide `2n` persons into `n` couples is given by:

\[ \text{Total ways} = \frac{2n!}{n! \cdot (2!)^n} \]

### Conclusion
Thus, the correct answer is option (c): \(\frac{2n!}{n!(2!)^n}\).

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