How many 4 digit numbers can be made by using 1, 2, 3, 4, 5, 6 & 7. Repetition…

Concept: When arranging r positions using items chosen without repetition from a pool of n distinct items, the Fundamental Counting Principle applies — multiply the number of choices available at each position, where the pool shrinks by one after each digit is placed. This is the same as the permutation count nPr = n! / (n − r)!.

Application:

1. There are 7 distinct digits available: 1, 2, 3, 4, 5, 6, 7.

2. The first position (thousands place) of the 4-digit number can be filled in 7 ways, since any of the 7 digits can be placed there.

3. Since repetition is not allowed, one digit is now used, so the second position can be filled in 6 ways from the remaining digits.

4. The third position can then be filled in 5 ways, and the fourth position in 4 ways, since each placement removes one more digit from the pool.

5. Multiplying the number of choices for all four positions gives the total count: 7 × 6 × 5 × 4 = 840.

Cross-check: This matches the permutation formula with n = 7 and r = 4: 7P4 = 7! / (7 − 4)! = 7! / 3! = 5040 / 6 = 840, confirming the count.

Hence, 840 four-digit numbers can be formed using the given digits without repetition.