The total number of ways of dividing `15` different things into groups of `8. ,4` and `3` respectively, is

Options

A. `(15!)/(8!4!(3!)^(2))`
B. `(15!)/(8!4!3!)`
C. `(15!)/(8!4!)`
D. none of these

Answer

To solve the problem of dividing 15 different things into groups of 8, 4, and 3 respectively, we can follow these steps:

### Step 1: Choose 8 items from 15
We start by selecting 8 items from the total of 15 items. The number of ways to choose 8 items from 15 is given by the combination formula:

\[ \binom{15}{8} = \frac{15!}{8! \times (15-8)!} = \frac{15!}{8! \times 7!} \]

### Step 2: Choose 4 items from the remaining 7
After selecting 8 items, we have 7 items left. Now, we need to choose 4 items from these remaining 7 items. The number of ways to choose 4 items from 7 is:

\[ \binom{7}{4} = \frac{7!}{4! \times (7-4)!} = \frac{7!}{4! \times 3!} \]

### Step 3: Assign the last 3 items
After selecting 8 and 4 items, we are left with 3 items. Since there is only one group left that requires exactly 3 items, there is only 1 way to assign these items to this group:

\[ \binom{3}{3} = 1 \]

### Step 4: Combine the results
Now, we multiply the number of ways to form each group together:

\[ \text{Total ways} = \binom{15}{8} \times \binom{7}{4} \times \binom{3}{3} \]

Substituting the values we derived:

\[ \text{Total ways} = \frac{15!}{8! \times 7!} \times \frac{7!}{4! \times 3!} \times 1 \]

### Step 5: Simplify the expression
Notice that the \(7!\) in the numerator and denominator cancels out:

\[ \text{Total ways} = \frac{15!}{8! \times 4! \times 3!} \]

### Final Result
Thus, the total number of ways of dividing 15 different things into groups of 8, 4, and 3 respectively is:

\[ \frac{15!}{8! \times 4! \times 3!} \]