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
