When product of r consecutive positive integers is divided by `r!` then the quotient is:

Options

A. any natural number
B. a perfect square
C. a perfect cube
D. a proper fraction

Answer

To solve the problem of finding the quotient when the product of `r` consecutive positive integers is divided by `r!`, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We need to find the quotient of the product of `r` consecutive positive integers divided by `r!`.

2. **Define the Product of `r` Consecutive Integers**:
The product of `r` consecutive positive integers starting from `n` can be expressed as:
\[ P = n \times (n + 1) \times (n + 2) \times \ldots \times (n + r - 1) \]

3. **Define `r!`**:
The factorial of `r`, denoted as `r!`, is defined as:
\[ r! = r \times (r - 1) \times (r - 2) \times \ldots \times 1 \]

4. **Set Up the Quotient**:
We need to compute the quotient:
\[ Q = \frac{P}{r!} \]

5. **Substituting Values**:
Let's substitute the product of `r` consecutive integers:
\[ Q = \frac{n \times (n + 1) \times (n + 2) \times \ldots \times (n + r - 1)}{r!} \]

6. **Example Calculation**:
To illustrate, let's take `r = 3`. The product of 3 consecutive integers starting from `n` would be:
\[ P = n \times (n + 1) \times (n + 2) \]
And `3!` is:
\[ 3! = 3 \times 2 \times 1 = 6 \]
Thus, the quotient becomes:
\[ Q = \frac{n \times (n + 1) \times (n + 2)}{6} \]

7. **Generalizing the Result**:
After performing the division, we can see that the quotient simplifies to a natural number, which is the result of dividing the product of the integers by the factorial.

8. **Conclusion**:
The quotient when the product of `r` consecutive positive integers is divided by `r!` is a natural number.