Question

The number of five-digit telephone numbers having at least one of their digits repeated is
(a) 90000
(b) 100000
(c) 30240
(d) 69760

Answer 1

(d) 69760
Total number of five digit numbers (since there is no restriction of the number 0XXXX) = 10$\times$10$\times$10$\times$10$\times$10 = 100000
These numbers also include the numbers where the digits are not being repeated. So, we need to subtract all such numbers.
Number of 5 digit numbers that can be formed without any repetition of digits = 10$\times$9$\times$8$\times$7$\times$6 = 30240
∴ Number of five-digit telephone numbers having at least one of their digits repeated = {Total number of 5 digit numbers} $-$ {Number of numbers that do not have any digit repeated} = 100000 $-$ 30240 = 69760