Question

How many 5 digit numbers can be formed using the digits 1,2,3,4,5,6,7,8,9 such that no two consecutive digits are the same?
A.None of these
B.$9 \times {8^4}$
C.${9^5}$
D.${8^5}$

Answer 1

Hint: To answer the permutation and combination question or to find the number of ways in which a 5 digit number can be arranged such that no two consecutive are together. So this problem filled the first digit of the five digit number in 9 ways and the rest of the boxes in 8 different ways.

Complete step-by-step answer:
Given the digits through which we have to form the five digit number such that the two consecutive numbers should not be together.

1st

2nd

3rd

4th

5th

So the 1st box can be formed using 9 different ways as there are no restrictions in the first digit.
The 2nd box can be filled in 8 different ways because a number is already filled in the first digit.
The 3rd box can be filled in 8 different ways because there are a total 9 numbers and 2nd box digits should not be consecutive numbers.
Similarly for 4th box, 8 different ways
5th box, 8 different ways
So the total number of ways in which 5 digit numbers using the digits 1,2,3,4,5,6,7,8,9
in which no two consecutive numbers should be together can be arranged in
Total number of possible numbers
⇒\[9 \times 8 \times 8 \times 8 \times 8 \Rightarrow 9 \times {\left( 8 \right)^4}\]
Hence the total number of ways is \[9 \times {\left( 8 \right)^4}\]
So, the correct answer is “Option B”.

Note: Students sometimes get confused and write the other answers. It may be due to the number of filling ways of the 2nd , 3rd , 4th and 5th box. These boxes can be filled by 8 different ways each.