Find the number of five-digit multiples of 5, where all the digits are different, and the second digit (from the left) is odd.
Accepted Solution
A:
Given: candidates for digits from left: 1. [1-9] (leftmost, 5 digit-number) 2. [13579] odd 3. [0-9] 4. [0-9] 5. [05] (multiple of 5).
We will start with digits with most constraints (5,2,1,3,4) For the fifth digit, number ends with either 0 or 5, so two cases.
Case 1: 5th digit is a 0 (one choice) 2nd digit can be chosen from [1,3,5,7,9] so 5 choices 1st digit can be chosen from [1-9] less second digit, so 8 choices 3rd digit can be chosen from remaining 7 choices 4th digit can be chosen from remaining 6 choices for a total of 1*5*8*7*6=1680 arrangements
Case 2 5th digit is a 5 (one choice) 2nd digit can be chosen from [1379] , so 4 choices 1st digit can be chosen from [1-4,6-9] less 2nd digit, so 8-1=7 choices 3rd digit can be chosen from remaining 7 choices 4th digit can be chosen from remaining 6 choices for a total of 1*4*7*7*6=1176
Case 1+ Case 2 = total number of arrangements for 5 digit multiples of 5 with second digit odd = 1680+1176 = 2856