How many 4 digit even numbers can be formed by using 1, 2, 3 and 4, if no one…

Concept: When counting arrangements under a positional restriction (such as forming an even number), apply the Fundamental Counting Principle — first count the choices available for the restricted position, then multiply by the number of ways to fill the remaining positions with the remaining items.

1. A 4-digit number formed from the digits 1, 2, 3 and 4 (each used exactly once) is even only when its units digit is even. Among 1, 2, 3, 4 there are exactly two even digits, so the units place can be filled in 2 ways.

2. Once the units digit is fixed, the remaining three digits must occupy the remaining three positions (thousands, hundreds, tens) in some order. Three distinct digits can be arranged in 3! = 3 x 2 x 1 = 6 ways.

3. By the Fundamental Counting Principle, multiply the number of independent choices: 2 x 6 = 12.

Cross-check: Of all 4! = 24 possible arrangements of the digits 1, 2, 3, 4, each of the 4 digits is equally likely to occupy the units place across the full set of arrangements. Since exactly 2 of the 4 digits are even, exactly half of the 24 arrangements end in an even digit: 24 / 2 = 12, which matches the result above.