Question

 How many numbers of 5 digit telephone numbers can be formed having at least one of their digits repeated?

Answer 1

Hint: The above problem can be solved by subtracting the number of 5 digit telephone numbers having not any one of their digits repeated from the total number of 5 digit telephone numbers are possible without any restriction.

Complete step by step answer:
To find the number of 5 digit telephone numbers having at least one of their digits repeated is to subtract the number of 5 digit telephone numbers with no digit repeated from the total number of 5 digit telephone numbers having any kind of digits possible.
We are going to write the total number of 5 digit telephone having any digits possible as follows:
5 digit telephone number means 5 single digits in a particular order. The digits in a telephone number can take value from 0 to 9 means total 10 digits are possible.
__ __ __ __ __
In the above 5 blanks, any one digit from 0 to 9 can be filled in all the blanks.
As, we are going to write the total number of 5 digit telephone numbers having any digits possible so first blank can take any number from 0 to 9 (10 possibilities). Similarly second blank can also take any number from 0 to 9, third, fourth and fifth blank also have 10 possibilities.
$\underline{10}\text{ }\underline{10}\text{ }\underline{10}\text{ }\underline{10}\text{ }\underline{10}$
The number shown in each blank corresponds to the total number of digits possible in each blank.
And the number of digits possible in one blank is independent of the other blank so the total number of 5 digit telephone numbers that are possible without any restriction are 10×10×10×10×10 which is equal to 105.
Now, we are going to write the number of 5 digit telephone numbers which are possible when no digit is repeated.
So, in a 5 digit telephone number all the digits of a number should be different.
$\underline{10}\text{ }\underline{9}\text{ }\underline{8}\text{ }\underline{7}\text{ }\underline{6}$
If the first blank can take any digit from 0 to 9 (10 possibilities) then the second blank cannot take the digit which is in the first blank so 9 possibilities are there. Similarly third blank has 8 possibilities, fourth blank has 7 possibilities and fifth blank has 6 possibilities.
So, the number of 5 digit telephone numbers which are possible when no digit is repeated is 10×9×8×7×6 which is equal to 30240.
Now, number of 5 digit telephone numbers can be formed having at least one of their digits repeated is equal to 69760 (=100000 – 30240),
Hence, the answer to the above problem is 69760.

Note: The number of ways 5 digit telephone numbers are possible without any digit is repeated is 10×9×8×7×6 which can also be thought of arranging 10 digits (0 to 9) in the 5 blanks such that no digit is repeated and can be written as${}_{5}^{10}P$.
$\begin{align}
  & {}_{5}^{10}P=\dfrac{10!}{\left( 10-5 \right)!} \\
 & \Rightarrow {}_{5}^{10}P=\dfrac{10\times 9\times 8\times 7\times 6\times 5!}{5!} \\
 & \Rightarrow {}_{5}^{10}P=10\times 9\times 8\times 7\times 6 \\
\end{align}$