Question

How many $3$ digit even numbers can be formed from the digits $1,2,3,4,5,6$ if the digits can be repeated ?

Answer 1

We have to find all the three digit even numbers which can be formed using the given digits. We solve this question using the concept of even numbers and the concept of combinations. We will solve it by making cases and possibilities of numbers formed using the given digits. First we will take three cases for the three digits separately and then multiply all the cases of the number of digits to find the total number of even numbers which can be formed using the given digits.

Complete step by step answer:
Given, $3$ digit even numbers can be formed from the digits $1,2,3,4,5,6$ if the digits can be repeated.
Now, we will take the number of possible digits which can satisfy the particular place as :
For the ones place:
As we are given that we have to form a three digit even number, this is only possible when a number ends with a digit which is a multiple of $2$.
From the given digits the even digits are: $2$ , $4$ , $6$.
So, the number of digits which can be placed at one's place is : $3$ .
For the tens and hundred place:
As we are given that the digits can be repeated , so all the given digits can be placed at the tens and hundreds place.
So, the number of digits which can be placed at tens and hundreds places are 6 (for both places) .
Now, for the total number which can be formed using the given digits can be calculated as :
$Total{\text{ }}numbers = \left( {number{\text{ }}of{\text{ }}ways{\text{ }}of{\text{ }}ones{\text{ }}place} \right) \times \left( {number{\text{ }}of{\text{ }}ways{\text{ }}of{\text{ }}tens{\text{ }}place} \right) \times \left( {number{\text{ }}of{\text{ }}ways{\text{ }}of{\text{ }}hundreds{\text{ }}place} \right)$
Now putting the values we get the value of total numbers as :
$Total{\text{ }}numbers = 3 \times 6 \times 6$
$Total{\text{ }}numbers = 108$

Hence, the total $3$ digit number which can be formed using the digits $1,2,3,4,5,6$ if the digits can be repeated are $108$ numbers.

Note:
We can also solve this question using the total number of three digit numbers and the number of odd three digit numbers by subtracting the number of three digit numbers from the total three digit numbers.