Find the number of five-digit multiples of 5, where all the digits are different, and the second digit (from the left) is odd.

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