From the digits {1, 2, 3, 4, 5, 5, 6, 6}, how many four-digit numbers can be…

Concept: When choosing r items from n available distinct values, where order matters and no value repeats within a selection, the count is the permutation nP r = n × (n − 1) × ... × (n − r + 1).

Application: The set {1, 2, 3, 4, 5, 5, 6, 6} contains only 6 distinct digit values — 1, 2, 3, 4, 5, 6 — since 5 and 6 each appear twice; a repeated card does not add a new distinct digit. Forming a 4-digit number with no digit repeated in the number is therefore choosing and arranging 4 of these 6 distinct digits.

1. Fill the 1st digit: any of the 6 distinct digits can go here — 6 ways.

2. Fill the 2nd digit: one digit is already used, so 5 distinct digits remain — 5 ways.

3. Fill the 3rd digit: 4 distinct digits remain — 4 ways.

4. Fill the 4th digit: 3 distinct digits remain — 3 ways.

5. Multiply the choices across the four positions: 6 × 5 × 4 × 3 = 360.

Cross-check: Using the permutation formula directly with n = 6, r = 4: 6P4 = 6! / (6 − 4)! = 720 / 2 = 360 — the same result, confirming the count.

Result: 360 four-digit numbers can be formed.