# The number of three digit numbers having no digits as $5$ is

$

{\text{A}}{\text{. 252}} \\

{\text{B}}{\text{. 225}} \\

{\text{C}}{\text{. 648}} \\

{\text{D}}{\text{. none of these}} \\

$

Answer

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Hint: This problem is based on the principle of fundamental counting which states that if there are n ways of doing something, and m ways of doing another thing after that, then there are $m \times n$ ways to perform both of these actions.

Complete step-by-step answer:

We know that

A three digit number has three places Ones, tenths and hundredths. Each place can be filled with any number between \[0{\text{ to 9}}\] .

For a three digit number without$5$ , hundreds place can be filled with any number except $0{\text{ and 5}}{\text{.}}$

Therefore the number of digits that can be placed at the hundredth’s place is $8$ .

Number 5 cannot be used as mentioned in the question.

So, number of digits that can be placed in tenth’s place is $9.$

And, number of digits that can be placed in one’s place is $9.$

So, from the principle of fundamental counting number of three digit numbers having no digits as $5$ is

$

= 8 \times 9 \times 9 \\

= 648 \\

$

Hence, the number of three digits numbers having no digits as \[5{\text{ is 648}}{\text{.}}\]

Note: This problem is based on fundamental counting principle and for similar problems like this we have to find the number of ways a task can be done. These types of problems exclude the way which is not required and count the others left. In a three digit number zero cannot be placed at the hundredth’s place because the number will no longer be a three digit number.

Complete step-by-step answer:

We know that

A three digit number has three places Ones, tenths and hundredths. Each place can be filled with any number between \[0{\text{ to 9}}\] .

For a three digit number without$5$ , hundreds place can be filled with any number except $0{\text{ and 5}}{\text{.}}$

Therefore the number of digits that can be placed at the hundredth’s place is $8$ .

Number 5 cannot be used as mentioned in the question.

So, number of digits that can be placed in tenth’s place is $9.$

And, number of digits that can be placed in one’s place is $9.$

So, from the principle of fundamental counting number of three digit numbers having no digits as $5$ is

$

= 8 \times 9 \times 9 \\

= 648 \\

$

Hence, the number of three digits numbers having no digits as \[5{\text{ is 648}}{\text{.}}\]

Note: This problem is based on fundamental counting principle and for similar problems like this we have to find the number of ways a task can be done. These types of problems exclude the way which is not required and count the others left. In a three digit number zero cannot be placed at the hundredth’s place because the number will no longer be a three digit number.

Last updated date: 20th Sep 2023

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