Question

How many 5-digit license plates can be make using the digits 1,2,3,4,5,6,7 if an odd digit must come first and repetitions are allowed? What if repetitions are not allowed

Answer 1

Repetitions allowed Answer: 9,604

Explanation:

First digit: {1, 3, 5, or 7} = 4 choices

2nd digit: {1, 2, 3, 4, 5, 6, or 7} = 7 choices

3rd digit: {1, 2, 3, 4, 5, 6, or 7} = 7 choices

4th digit: {1, 2, 3, 4, 5, 6, or 7} = 7 choices

5th digit: {1, 2, 3, 4, 5, 6, or 7} = 7 choices

1st digit and 2nd digit and 3rd digit and 4th digit and 5th digit

4 x 7 x 7 x 7 x 7

= 4(7)⁴

= 9,604

*******************************************************************************

Repetitions not allowed Answer: 1,440

Explanation:

First digit: {1, 3, 5, or 7} = 4 choices

2nd digit: {1, 2, 3, 4, 5, 6, or 7} - {1st digit} = 7 - 1 = 6 choices

3rd digit: {1, 2, 3, 4, 5, 6, or 7} - (1st & 2nd digits} = 7-2 = 5 choices

4th digit: {1, 2, 3, 4, 5, 6, or 7} - {1st, 2nd, & 3rd digits} = 7-3 = 4 choices

5th digit: {1, 2, 3, 4, 5, 6, or 7} - (1st, 2nd, 3rd, & 4th digits} = 7-4 = 3 choices

1st digit and 2nd digit and 3rd digit and 4th digit and 5th digit

4 x 6 x 5 x 4 x 3

= 1,440