Question

Find the number of 10 digits numbers that can be formed using the digits 0, 1, 2, 3………..9 without repetition.

Answer 1

Hint: Notice that the number of digits given is 10 and we are asked the number of ten-digit numbers that can be formed using these digits, so the number of ten-digit numbers that can be formed is equal to the number of arrangements of the digits possible except the arrangements in which 0 appears at the first place.

Complete step-by-step solution -
As the number of digits given is 10 and we are asked the number of ten-digit numbers that can be formed using these digits, so the number of ten-digit numbers that can be formed is equal to the number of arrangements of the digits possible except the arrangements in which 0 appears at the first place, as when 0 will appear in the first place the number would be a 9 digit number. So, first we will find all the possible arrangements of the digits given in the question and subtract the cases when zero appeared in the first place.
Therefore, the number of possible arrangements of the given digits is $^{10}{{P}_{10}}$ , which is equal to $10!$.
Now see if we fix the position of 0. We are left with 9 digits and 9 places to arrange them. So, the number of arrangements in which 0 appears in the first place is $^{9}{{P}_{9}}$ , which is equal to 9!.
Therefore, the number of 10 digit numbers that can be formed using the digits 0, 1, 2, 3………..9 without repetition is $(10!-9!)$ which is equal to $9!(10-1)=9.9!$.

Note: In the question related to permutations and combinations, students are generally confused if the question is based on arrangement or selection. Also, in questions like the above one students generally miss out on the step of subtracting the cases in which 0 appears in the first place.