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

Answer

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

### Step 1: Understand the distribution
We need to distribute 12 different items among 5 people with the following distribution:
- Person A: 2 items
- Person B: 2 items
- Person C: 2 items
- Person D: 3 items
- Person E: 3 items

### Step 2: Calculate the total arrangements
The total number of arrangements can be calculated using the formula for permutations of a multiset:
\[ \text{Total arrangements} = \frac{n!}{n_1! \times n_2! \times n_3! \times n_4! \times n_5!} \]
where \( n \) is the total number of items, and \( n_1, n_2, n_3, n_4, n_5 \) are the counts of items assigned to each person.

### Step 3: Substitute the values
In our case:
- \( n = 12 \) (total items)
- \( n_1 = 2 \) (for person A)
- \( n_2 = 2 \) (for person B)
- \( n_3 = 2 \) (for person C)
- \( n_4 = 3 \) (for person D)
- \( n_5 = 3 \) (for person E)

Thus, we can substitute these values into the formula:
\[ \text{Total arrangements} = \frac{12!}{2! \times 2! \times 2! \times 3! \times 3!} \]

### Step 4: Simplify the expression
Now we can simplify the expression:
\[ = \frac{12!}{(2!)^3 \times (3!)^2} \]

### Step 5: Calculate the factorials
Now we can calculate the factorials:
- \( 2! = 2 \)
- \( 3! = 6 \)

Thus:
\[ (2!)^3 = 2^3 = 8 \]
\[ (3!)^2 = 6^2 = 36 \]

### Step 6: Substitute back into the equation
Now substituting back:
\[ = \frac{12!}{8 \times 36} \]
\[ = \frac{12!}{288} \]

### Step 7: Final calculation
Now we can compute \( 12! \) and divide it by 288 to get the final answer.

### Final Answer
The number of ways to distribute the items is:
\[ \frac{479001600}{288} = 1663200 \]

Thus, the total number of ways to divide the 12 different things among 5 persons so that they can get 2, 2, 2, 3, and 3 items respectively is **1663200**.

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