Identify Single-Digit Prime Numbers

First, list the single-digit prime numbers. Prime numbers are greater than 1 and have only two divisors: 1 and themselves. The single-digit primes are:

• 2
• 3
• 5
• 7

There are 4 distinct single-digit prime numbers available.

Calculate Number of 3-Digit Numbers

We need to form 3-digit numbers using three distinct digits from the set {2, 3, 5, 7}. Since the digits must be distinct and cannot include 0 (as 0 is not a prime digit), the order of the digits matters, and we are essentially finding the number of permutations.

We need to choose and arrange 3 digits out of the 4 available prime digits. This can be calculated using the permutation formula:

$ P(n, k) = \frac{n!}{(n-k)!} $

Where:

• $n$ is the total number of items to choose from (4 single-digit primes).
• $k$ is the number of items to choose and arrange (3 digits for the number).

Substituting the values:

$ P(4, 3) = \frac{4!}{(4-3)!} = \frac{4!}{1!} $

Calculate the factorial:

$ 4! = 4 \times 3 \times 2 \times 1 = 24 $

$ 1! = 1 $

Therefore:

$ P(4, 3) = \frac{24}{1} = 24 $

Alternatively, consider the positions:

• For the first digit (hundreds place), there are 4 choices.
• For the second digit (tens place), there are 3 remaining choices (since digits must be distinct).
• For the third digit (units place), there are 2 remaining choices.

Total number of 3-digit numbers = $4 \times 3 \times 2 = 24$.

Thus, there are 24 possible 3-digit numbers that can be formed using three distinct single-digit prime numbers.