How many 4-digit numbers can be formed using the digits 0, 1, 2, 3, 4 without…

CONCEPT

For a number formed without repetition, choices are counted position by position using the multiplication principle.

A leading zero is not allowed in a multi-digit number, so the first position must be counted separately from the remaining positions.

APPLICATION

1. The thousands place has 4 choices: 1, 2, 3, or 4.

2. After choosing the first digit, 4 digits remain for the hundreds place.

3. Then 3 digits remain for the tens place and 2 digits remain for the units place.

4. By the multiplication principle, the total is 4 × 4 × 3 × 2 = 96.

CROSS-CHECK / CONTRAST

Equivalently, all 5P4 arrangements total 120, but 0-leading arrangements are 4P3 = 24; hence 120 − 24 = 96.

• 120 counts all arrangements of four distinct digits chosen from five, including the arrangements that begin with 0.

• 54 permits repetition because it gives all five digits as choices at each of four positions.

• 45 models five positions with four choices each, which does not match the given four-digit construction.

• The compound threshold-two alternative does not apply because none of the numerical expressions equals 96.

• Because 96 is absent from all displayed numerical expressions, the universal-exclusion alternative applies.

Therefore, 96 such numbers can be formed, and the universal-exclusion alternative is selected.