Question

Assuming that no two consecutive digits are same, the number of n digit numbers is
1) n!
2) 9!
3) $9^n$
4) $n^9$

Answer 1

Hint: We are to find the number of n-digit numbers where no two consecutive digits are the same. We know that a number system has a total of 10 digits (0 to 9). Then, by using permutation and combination, we will check the number of ways each place can be filled. And depending on this, we will see the nth place we can fill in how many ways. By doing so, we will get the final output.

Complete step-by-step answer:
As we know that, there are a total 10 digits i.e. 0 to 9 in the number system.
Also, given that, there are n-digits in the given number.
So, here the first place can have any of the 9 digits except 0.
This means that the first place can be filled in 9 ways.
Also, the second place can have any of the 9 digits except the digit which is at the first place.
This means that the second place can be filled in 9 ways.
Similarly the third, fourth,…,nth place can have any of the 9 digits.
This means that all the places can be filled in 9 ways.
Thus, the total number of ways to fill the nth place
\[ = 9 \times 9 \times ........ \times 9 \times 9\] (n times)
\[ = {9^n}\]
Hence, number of n-digit numbers, with no two consecutive digits same is given by \[{9^n}\]
So, the correct answer is “Option 3”.

Note: Another Method:
Given that, no two consecutive digits are the same.
This means that there are n numbers starting from 1 and not 0 in the first place.
Let there be 1 to 9 numbers.
So, the number of ways to fill the first place of the n-digit number = 9 ways.
Now, here we are given that no two consecutive numbers are the same.
So, if we keep number 2 in the first place, then we can write numbers 0, 1,…, 9 from second place onwards.
Thus, the second place of the n-digit number can be filled in 9 ways.
Hence, the total number of ways to fill first 2-places of n-digit number
\[ = 9 \times 9\]
\[ = 81\]
\[ = {9^2}\]
Similarly, total number of ways to fill n-places of n-digit number is
\[ = {9^n}\]