Question

How many 8 digit mobile numbers can be formed if at least one digit is repeated?
A) ${10^8}$
B) ${10^7}$
C) ${10^8} - {}^{10}{{\rm{P}}_7}$
D) ${10^8} - {}^{10}{{\rm{P}}_8}$

Answer 1

Here we have to use the concept of permutation to calculate the total possible 8 digit mobile number can be formed if at least one digit is repeated. So, firstly we will calculate the total number that can be formed if any digit can be repeated and then we will find out the number that can be formed if no digit is repeated. The difference between them will give us the numbers that are formed if at least one digit is repeated.

Complete step by step solution:
So, firstly we have to calculate the total numbers that can be formed if any digit can be repeated. Hence, any of 10 digits i.e. $0,1,2,3,4,5,6,7,8,9$ are available to be placed at each place of the 8 digit number. Hence, numbers of 8 digit mobile number that can be formed if any digit can be repeated $= 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10$
Therefore, numbers of 8 digit mobile number that can be formed if any digit can be repeated $= {10^8}$… (1)
Now we have to find out the numbers that can be formed if no digit is repeated. Hence, for ${1^{{\rm{st}}}}$ position in the 8 digit mobile number any of the 10 digits can be placed and for ${2^{{\rm{nd}}}}$ position any of the remaining 9 digits can be placed and for ${3^{{\rm{rd}}}}$ position any of the remaining 8 digits can be placed and so on. Hence, numbers of 8 digit mobile number that can be formed if no digit is repeated $= 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 = {}^{10}{{\rm{P}}_8}$
Therefore, numbers of 8 digit mobile number that can be formed if no digit is repeated $= {}^{10}{{\rm{P}}_8}$… (2)
Now, numbers of 8 digit mobile number that are formed if at least one digit is repeated is equal to the difference between numbers of 8 digit mobile number that can be formed if any digit can be repeated and numbers of 8 digit mobile number that can be formed if no digit is repeated i.e. difference between equation (1) and equation (2)
Hence, numbers of 8 digit mobile number that are formed if at least one digit is repeated $= {10^8} - {}^{10}{{\rm{P}}_8}$

Therefore, option D is the correct answer.

Note:
Permutations may be defined as the different ways in which a collection of items can be arranged. For example: The different ways in which the numbers 1, 2 and 3 can be grouped together, taken all at a time, are$123,{\rm{ }}132,{\rm{ }}213{\rm{ }}231,{\rm{ }}312,{\rm{ }}321$.
So, Number of permutations of n things, taken r at a time, denoted by ${}^{\rm{n}}{{\rm{P}}_{\rm{r}}} = \dfrac{{{\rm{n}}!}}{{{\rm{(n - r)}}!}}$
Combinations may be defined as the various ways in which objects from a set may be selected. For example: The different selections possible from the numbers 1, 2, 3 taking 2 at a time, are ${\rm{12, 23 and 31}}$
So, Number of combinations possible from n group of items, taken r at a time, denoted by $^{\rm{n}}{{\rm{C}}_{\rm{r}}} = \dfrac{{{\rm{n}}!}}{{{\rm{r}}!{\rm{(n - r)}}!}}$