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(D) The total number of $5$ digit telephone numbers that can be formed using digits $0, 1, 2, \dots, 9$ is $10^5 = 100000$ (since each position can be

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$69760$

Vedclass · Answer

(D) The total number of $5$ digit telephone numbers that can be formed using digits $0, 1, 2, \dots, 9$ is $10^5 = 100000$ (since each position can be filled in $10$ ways).The number of $5$ digit telephone numbers with no digits repeated is given by the permutation formula $^{10}P_5 = \frac{10!}{(10-5)!} = 10 	imes 9 	imes 8 	imes 7 	imes 6 = 30240$.Therefore,the number of telephone numbers having at least one digit repeated is the total number of telephone numbers minus the number of telephone numbers with no digits repeated.Required number $= 100000 - 30240 = 69760$.

The number of $5$ digit telephone numbers having at least one of their digits repeated is

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