Question

The digits 1 to 7 are used to create a four digit code to enter a locked room.How many different process are possible if the digits may not be repeated and the code must be an even number bigger than 5000.?Please explain

Answer 1

Answer: 160

Solution:

First digit can be: 5, 6, 7 (3 variants totally).

Second digit can be: any (7 variants totally).

Third digit can be: any (7 variants totally).

Fourth digit can be: 2,4,6 (3 variants totally).

If the first digit is 5, than there are 3 ways to choose the fourth digit, 5 ways to choose the third digit, 4 ways to choose the second digit.

If the first digit is 6, than there are 2 ways to choose the fourth digit, 5 ways to choose the third digit, 4 ways to choose the second digit.

If the first digit is 7, than there are 3 ways to choose the fourth digit, 5 ways to choose the third digit, 4 ways to choose the second digit.

Overall:

$Number=3×5×4\,+\,2×5×4\,+\,3×5×4=60+40+60=160$