Question

Find the total number of 9 digit numbers which will have all different digits.
A. $9!$
B. $8!$
C. $9 \times 9!$
D. None of these

Answer 1

Hint: We will first notice that we can use 10 digits to fill up 9 spaces but 0 cannot take the first place. Thus, after that, we will write the number of possible digits which can take place in that space and multiply all the possibilities to get the final answer.

Complete step-by-step answer:
We know that we have 10 nine digit numbers with us which are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
Now, we need to find 9 digit numbers.
So, let us consider we have 9 blank spaces to fill in order to get a 9 digit number.
We have _ _ _ _ _ _ _ _ _.
We now just need to find the number of possible digits we can fill in each space.
Let us say we can have 10 numbers in first place, but if we put in 0, we will get an eight digit number.
Hence, the first space can be filled with 9 digits.
Now, the second space can take all 10 digits, but we need numbers with no repetition. So, we see 1 number has already been booked by first digit. So, we now have 9 digits to fill up the second place. Now, after that 8 digits for third place, 7 digits for fourth place, 6 digits for fifth place, 5 digits for sixth place, 4 digits for seventh place, 3 digits for eighth place and finally 2 digits for ninth place.
So, the number of ways will be = $9 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2$.
Multiplying by 1 would not change anything. So, we can rewrite it as:
Number of ways are = $9 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$ ……….(1)
We know that $n! = n \times (n - 1) \times (n - 2)........1$
Using this in (1), we will have:-
\[9 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 9 \times 9!\].

So, the correct answer is “Option C”.

Note: The student may forget that he/she needs the number of nine digit numbers with no digits being repeated. They will then take the values as 9 digits in first place and then 10 in every other place, which will obviously change our answers.
The students might take the first digit options as 10, because they may forget that it will make the numbers with 8 digits as well.