Answer:
a) 625,000
b) 78,125 if 0 is counted as even or 40,000 if it is not.
Explanation:
Represent the 7 digit number by abcdefg
Position Allowable digits Number of allowable digits
g 1,3,5,7,9 5
f 0,1,2,3,4,5,6,7,8,9 10
e 1,3,5,7,9 5
d 0,1,2,3,4,5,6,7,8,9 10
c 1,3,5,7,9 5
b 0,1,2,3,4,5,6,7,8,9 10
a 1,3,5,7,9 5
This implies that the number of 7 digit numbers with odd digits in the odd positions is
5x10x5x10x5x10x5=625000
For part b, count 0 as an even digit. Then rule out 1,3,5,7,9 in positions b, d, and f. This leaves 5 possible digits in all 7 locations. So the answer is
5^7=78,125
If 0 is not counted as an even digit, then the answer is
4^3+5^4=40000