Counting prinicples

Question

The digits 1 to 7 are used to create a four digit code to enter a locked room.How many different codes are possible if the digits may not be repeated and the code must be an even number bigger than 5000.Please explain to me how to do it?I have no idea what’s going on,

Answer 1

There are 3 possible choices for the first digit (5, 6 and 7) since the number must be >5000.

There are 2 possible choices for the last digit (2, 4 and 6) since the number must be even.

So our numbers will be of one of the following forms:

5 __ __ 2

5 __ __ 4

5 __ __ 6

6 __ __ 2

6 __ __ 4

(note that 6 __ __ 6 doesn't work since it has two digits of 6).

7 __ __ 2

7 __ __ 4

7 __ __ 6

So there are 8 combinations for the first and last digits.

We then have 5 choices for the second digit (in the case of 5 __ __ 2, the second digit can be any of 1, 3, 4, 6 or 7) and then once the second digit is chosen, we have four choices for the third digit (same options as the second digit except for the value chosen as the second digit). This gives us 5 x 4 = 20 choices for each combination of first and last digits, which means there are a total of 8 x 20 = 160 possible codes.