In how many ways can a committee schedule three speakers for the different meetings if they are all available on any of five possible dates ?

Options

A. 10
B. 36
C. 60
D. 120

Answer

To solve the problem of scheduling three speakers for different meetings on any of five possible dates, we can use the concept of permutations since the order of speakers matters.

### Step-by-Step Solution:

1. **Identify the Number of Speakers and Dates:**
- We have 3 speakers and 5 possible dates.

2. **Determine the Number of Ways to Choose Speakers:**
- We need to choose 3 speakers from the available 5 dates. Since the order matters (as they are scheduled for different meetings), we will use permutations.
- The formula for permutations is given by:
\[ P(n, r) = \frac{n!}{(n - r)!} \]
- Here, \( n = 5 \) (the number of dates) and \( r = 3 \) (the number of speakers).

3. **Calculate the Permutations:**
- Plugging the values into the formula:
\[ P(5, 3) = \frac{5!}{(5 - 3)!} = \frac{5!}{2!} \]

4. **Simplify the Factorials:**
- We can expand \( 5! \) as follows:
\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 \]
- And \( 2! = 2 \times 1 \).
- Therefore:
\[ P(5, 3) = \frac{5 \times 4 \times 3 \times 2!}{2!} \]
- The \( 2! \) cancels out:
\[ P(5, 3) = 5 \times 4 \times 3 \]

5. **Calculate the Final Result:**
- Now, compute the multiplication:
\[ 5 \times 4 = 20 \]
\[ 20 \times 3 = 60 \]

6. **Conclusion:**
- The total number of ways to schedule 3 speakers for different meetings on any of the 5 possible dates is **60**.