Question

The number of three-digit numbers having only two consecutive digits identical is

• A. $153$
• B. $162$
• C. $180$
• D. $161$

Answer 1

Answer: (B)

Let us consider two cases which will occur according to the given question.
Case $1$ : First two digits will be identical
Case $2$ : Last two digits will be identical
Take case $1$ ,
Number of ways we can select a digit for hundredth place $\to$ $9$ (Given that it is a three-digit number so $0$ cannot occur at this place)
Number of ways we can select a digit for ten's place $\to$ $1$ ( $\because$ It has to be same as the digit at hundredth place)
Number of ways we can select a digit for unit place $\to$ $9$ ( $\because$ It is different from digit at tenth place)
$\therefore$ Total number of three-digit numbers $=9\times 1\times 9=81$ .

Take case $2$ ,
Number of ways we can select a digit for hundredth place $\to$ $9$ (Given that it is a three-digit number so $0$ cannot occur at this place)
Number of ways we can select a digit for ten's place $\to$ $9$ ( $\because$ It is different from digit at hundredth place)
Number of ways we can select a digit for unit place $\to$ $1$ ( $\because$ It has to be same as the digit at tenth place)
$\therefore$ Total number of three-digit numbers $=9\times 9\times 1=81$ .
Number of three-digit numbers having only two consecutive digits identical is $=81+81=162$ .