Number of ways in which 12 different things can be divided among five persons so that they can get 2,2,2,3,3 things respectively is

Options

A. `(12!)/((3!)^(2)(2!)^(3))`
B. `(12!5!)/((3!)^(2)(2!)^(3))`
C. `(12!)/((3!)^(2)(2!)^(4))`
D. `(12!5!)/((3!)^(2)(2!)^(4))`

Answer

To solve the problem of distributing 12 different things among 5 persons such that they receive 2, 2, 2, 3, and 3 items respectively, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the total items and their distribution**: We have 12 different items and we need to distribute them among 5 persons such that three persons receive 2 items each and two persons receive 3 items each.

2. **Calculate the total arrangements of items**: Since all 12 items are different, the total number of ways to arrange these items is given by \(12!\) (12 factorial).

\[ \text{Total arrangements} = 12! \]

3. **Account for identical distributions**: Since three persons receive 2 items each, we need to divide by the arrangements of these identical distributions. The number of ways to arrange 3 persons receiving 2 items each is given by \(3!\) (3 factorial) because they are indistinguishable in terms of the number of items they receive.

\[ \text{Ways for 2 items} = 3! = 6 \]

4. **Account for the other identical distributions**: Similarly, for the two persons who receive 3 items each, we also need to divide by \(2!\) (2 factorial) because they are indistinguishable as well.

\[ \text{Ways for 3 items} = 2! = 2 \]

5. **Combine the calculations**: Now, we combine these calculations to find the total number of ways to distribute the items.

\[ \text{Total ways} = \frac{12!}{(2!)^3 \times (3!)^2} \]

Here, \((2!)^3\) accounts for the three persons receiving 2 items each, and \((3!)^2\) accounts for the two persons receiving 3 items each.

6. **Final expression**: Thus, the final expression for the number of ways to distribute the items is:

\[ \text{Total ways} = \frac{12!}{2^3 \times 3^2} \]

### Final Calculation:
- Calculate \(12!\), \(2^3\), and \(3^2\):
- \(12! = 479001600\)
- \(2^3 = 8\)
- \(3^2 = 9\)

- Now substitute these values into the expression:

\[ \text{Total ways} = \frac{479001600}{8 \times 9} = \frac{479001600}{72} = 6652800 \]

### Conclusion:
The total number of ways to distribute the 12 different items among the 5 persons such that they receive 2, 2, 2, 3, and 3 items respectively is **6652800**.