Question

how many integers from 0 through 999,999 contain the digit 4 exactly twice?

how many integers from 1 through 1000000 contain the digits 6 at least once

Answer 1

12. a. Integers between 0 and 999999 can be represented as below. First reserve 2 places for digit 4. So, select 2 places out of 6 in ()15 ways and place digit 4 in 1 way. The remaining places can fill by any digit except digit 4. So, each of remaining 4 placed can be filled in 9 ways. So, the number of integers between 0 and 999999 that contains the digit 4 exactly twice is 15-1.9- 98415 b. Integers between 1 and 999999 can be represented as below. Since 1000000 does not contains digit 6. So, 1000000 will not be included anyway. 6! 2 214 First reserve 1 place for digit 6 which will ensure that the integer contains at least one digit 6. So, select 1 place out of 6 in 6 ways and place digit 6 in way The remaining places can fill by any digit. So, each of remaining 5 placed can be filled in 10 ways. So, the number of integers between 1 and 1000000 that contains the digit 6 at least once is 6-1-105 600000