How many odd numbers between 10 and 99 have distinct digitsS

How many odd numbers between 10 and 99 have distinct digits?

Solution

Sol)

A number a₁a₂ between 10 and 99 can be viewed as an ordered tuple (a1, a2). Since a1a2 10 and a1a2 is odd, then a1 = 1, 2, . . . , 9 and a2 = 1, 3, 5, 7, 9. Since a1, a2 are distinct, we conclude: a2 has 5 choices; when a2 is fixed, a1 has 8(= 9 1) choices; Thus the answer is 8 × 5 = 40.