# The number of five-digit telephone numbers having at least one of their digits repeated is

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

Solution:

(d) 69760

Total number of five digit numbers (since there is no restriction of the number $0 X X X X)=10 \times 10 \times 10 \times 10 \times 10=100000$

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 $=10 \times 9 \times 8 \times 7 \times 6=30240$

$\therefore$ 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 $\}=100000-30240=69760$