Question

How many 9-digit mobile numbers can be formed if a digit can be represented any number of times?
A) ${}^{10}{{P}_{9}}$
B) $9\times 1{{0}^{8}}$
C) $^{10}{{C}_{9}}$
D) $1{{0}^{10}}$

Answer 1

Hint: First of all, we will keep 9 blanks and then one by one we will keep all the numbers in first place and find the permutations that can be formed and then at last we will add all of them to find the final solution.
Complete step-by-step answer:
Now, as given in the question we need to find how many 9-digits mobile numbers can be formed with repetition and for that we will assume digits 0-9 as 10 different digits.
Now, we will make 9 blanks in which we will keep all 10 digits one by one as fixed digits as given below:
$\_\_,\ \_\_,\_\_,\ \_\_,\_\_,\ \_\_,\_\_,\ \_\_,\_\_$
Now, we will consider 10 digits i.e. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 through which we will make possible a 9 digit number. Now, we know that the first place cannot be used by 0 because by doing that the number will become of 8 digits so it can be said that the first blank can be filled in 9 different ways i.e. by keeping 1, 2, 3, 4, 5, 6, 7, 8, or 9 fixed in first blank. So, the blanks will now look as.
$9,\ \_\_,\_\_,\ \_\_,\_\_,\ \_\_,\_\_,\ \_\_,\_\_$
Now, let's assume that we have used 1 in the first blank then, we will have 0, 2, 3, 4, 5, 6, 7, 8 and 9 as remaining digits. But, in question it is mentioned that we can use the same digits, so again we will have all the 10 digits for the second blank. Thus, the second blank will look as,
$9,\ 10,\_\_,\ \_\_,\_\_,\ \_\_,\_\_,\ \_\_,\_\_$
In the same way we can solve the rest of the 7 blanks and all the blanks will have 10 different ways of writing the digits.
Finally, the number of ways of making a 9-digit number can be given by
$9\times 10\times 10\times 10\times 10\times 10\times 10\times 10$
The above equation can also be written as,
$9\times {{\left( 10 \right)}^{8}}$.
Hence, $9\times {{\left( 10 \right)}^{8}}$ different mobile numbers of 9-digit can be formed and represented.
Thus, option (B) is correct.

Note: While solving such type of questions, students must read the question carefully because if in the question it is given that we have to use distinct numbers or the numbers can not be repeated then the answer will change i.e. $9\times 9\times 8\times 7\times 6\times 5\times 4\times 3$ which is wrong. So, students should keep this in mind and solve the sums accordingly.