The number of ways in which `p+q` things can be divided into two groups containing `p and q` things respectively is

Options

A. `((p+q+r)!)/(p! q! r!)`
B. `((pqr)!)/((p+q+r)!)`
C. `((p+q+r)!)/((pqr)!)`
D. None of these

Answer

To find the number of ways in which \( p + q \) things can be divided into two groups containing \( p \) and \( q \) things respectively, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**: We have a total of \( p + q \) items, and we need to divide them into two groups: one group containing \( p \) items and the other group containing \( q \) items.

2. **Choosing Items for Group 1**: We need to select \( p \) items from the total \( p + q \) items to form Group 1. The number of ways to choose \( p \) items from \( p + q \) items is given by the binomial coefficient:
\[ \binom{p+q}{p} \]

3. **Understanding the Remaining Items**: Once we have chosen \( p \) items for Group 1, the remaining \( q \) items will automatically go into Group 2. There is only one way to assign these remaining items to Group 2.

4. **Using the Binomial Coefficient Formula**: The binomial coefficient \( \binom{n}{r} \) is calculated as:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
Applying this to our case, we have:
\[ \binom{p+q}{p} = \frac{(p+q)!}{p! \cdot q!} \]

5. **Final Result**: Therefore, the total number of ways to divide \( p + q \) items into two groups of \( p \) and \( q \) items respectively is:
\[ \frac{(p+q)!}{p! \cdot q!} \]

### Final Answer:
The number of ways to divide \( p + q \) things into two groups containing \( p \) and \( q \) things respectively is:
\[ \frac{(p+q)!}{p! \cdot q!} \]