Question

A four or five-digit pin is to be selected from the seven digits 0,1,2,3,4,5,6. How many 5-digit pins are possible if:

1) all 5 digits are the same
2) no digit can be repeated
3) any digit can be repeated
4) the first 3 digits must be different digits and the last 2 digits must be the same, but none of the first 3 digits can be used in the last 2

Answer 1

For a 5-digit pin using the digits 0,1,2,3,4,5,6: 1) If all digits are the same, there is only one possible pin. 2) If no digit can be repeated, there are 5040 possible pins. 3) If any digit can be repeated, there are 16,807 possible pins. 4) If the first 3 digits must be different and the last 2 digits must be the same, and none of the first 3 digits can be used in the last 2, there are 1,260 possible pins.

Step-by-step explanation:

1) If all 5 digits are the same, there is only one possible 5-digit pin, as all digits must be the same.

2) If no digit can be repeated, there are 7 choices for the first digit, 6 choices for the second digit, 5 choices for the third digit, 4 choices for the fourth digit, and 3 choices for the fifth digit. This gives us a total of 7*6*5*4*3 = 5040 possible 5-digit pins with no repeated digits.

3) If any digit can be repeated, there are 7 choices for each of the 5 digits. This gives us a total of 7^5 = 16,807 possible 5-digit pins with repeated digits.

4) If the first 3 digits must be different and the last 2 digits must be the same, and none of the first 3 digits can be used in the last 2, there are 7 choices for the first digit, 6 choices for the second digit, 5 choices for the third digit, 6 choices for the fourth digit (any digit except the first 3), and 1 choice for the fifth digit (it must be the same as the fourth digit).

This gives us a total of 7*6*5*6*1 = 1,260 possible 5-digit pins with these conditions.