5-digit numbers from {3, 4, 5, 6, 7} with repetition allowed: How many distinct 5-digit numbers can be formed using the digits 3, 4, 5, 6, 7 when repetition is allowed?

Introduction / Context:Each of the 5 positions can independently take any of the five allowed digits when repetition is allowed. The multiplication principle directly gives the count as 5^5.

Given Data / Assumptions:

• Digits available: 5 (3, 4, 5, 6, 7).
• Number length: 5 digits.
• Repetition allowed; leading digit can be any of the five.

Concept / Approach:

• Choices per position = 5; positions = 5.
• Total = 5^5.

Step-by-Step Solution:

Verification / Alternative check:Counting by cases is unnecessary because every position has identical freedom (no restrictions on leading digit among the given five).

Why Other Options Are Wrong:

• 625 = 5^4 (only 4 positions).
• 125 = 5^3 (only 3 positions).
• 3905 is not a power of 5 and does not arise here.

Common Pitfalls:

• Accidentally disallowing repeats or restricting the leading digit.

Final Answer:3125