Question

How many odd numbers between 1000 and 9999 have distinct digits?

Answer 1

A number a1a2a3a4 between 1000 and 9999 can be viewed as an ordered tuple (a1, a2, a3, a4). Since a1a2a3a4 ≥

A number a1a2a3a4 between 1000 and 9999 can be viewed as an ordered tuple (a1, a2, a3, a4). Since a1a2a3a4 ≥ 1000 and a1a2a3a4 is odd, then a1 = 1, 2, . . . , 9 and a4 = 1, 3, 5, 7, 9. Since a1, a2, a3, a4 are distinct, we conclude:

A number a1a2a3a4 between 1000 and 9999 can be viewed as an ordered tuple (a1, a2, a3, a4). Since a1a2a3a4 ≥ 1000 and a1a2a3a4 is odd, then a1 = 1, 2, . . . , 9 and a4 = 1, 3, 5, 7, 9. Since a1, a2, a3, a4 are distinct, we conclude: a4 has 5 choices; when a4 is fixed, a1 has 8(= 9 − 1) choices; when a1 and a4 are fixed, a2 has 8(= 10 − 2) choices;

A number a1a2a3a4 between 1000 and 9999 can be viewed as an ordered tuple (a1, a2, a3, a4). Since a1a2a3a4 ≥ 1000 and a1a2a3a4 is odd, then a1 = 1, 2, . . . , 9 and a4 = 1, 3, 5, 7, 9. Since a1, a2, a3, a4 are distinct, we conclude: a4 has 5 choices; when a4 is fixed, a1 has 8(= 9 − 1) choices; when a1 and a4 are fixed, a2 has 8(= 10 − 2) choices; and when a1, a2, a4 are fixed, a3 has 7(= 10 − 3) choices. Thus the answer is 8 × 8 × 7 × 5 = 2240.