How many positive integers less than 1000?
a) have distinct digits
b) have distinct digits and are even
c) divisible by 7
d) divisible by 7 and not 11
c) both 7 and 11
d) either 7 or 11
e) exactly one of 7 or 11
f) neither 7 or 11
Please see the detailed solution in the attached file.
a)have distinct digits
Answer: There are 9 one-digital positive numbers. For two-digit numbers, the first digit may be anything from 1 to 9, while the second may be anything from 0 to 9 except the first digit. We thus have 9 choices for each digit, and so have 81 two-digit numbers with distinct digits. For three-digit numbers, we have 9 choices for the first digit, 9 for the second, and 8 for the third, for a total of 648 (9*9*8=648) three-digit numbers with distinct digits. Thus, there are 9+81+648=738 numbers less than ...
The solution is comprised of detailed explanations on how to find the number of postive integers less than 1000 with various restrictions, such as distinct digits, divisible by 7.