Question

Find the number of $4$-digit even numbers that can be formed using the digits $1,2,3,4,5$ if no digit is repeated. How many of those will be even?

Answer 1

(i)A four digit number is to be formed from the digits 1,2,3,4,5.
$\square \square \square \square T_{h}HTO$
Now, ones place can be filled in 5 ways.
Since, repetition is not allowed, tens place can be filled by remaining 4 digits.
So, there are 4 ways to fill ten's place.
Since, repetition is not allowed , so hundreds place can be filled by remaining 3 digits .
So, hundred's place can be filled in 3 ways.
Similarly, to fill thousand's place, we have 2 digits remaining.
So,thousand's place can be filled in 2 ways.
So, required number of ways in which four digit numbers can be formed from the given digits is $5\times 4\times 3\times 2=120$
(ii) Now, for the number to be even , ones place can be filled by 2 or 4.
So, there are 2 ways to fill one's place.
Since, repetition is not allowed, so tens place can be filled by remaining four digits.
So, tens place can be filled in 4 ways.
Since, repetition is not allowed , so hundreds place can be filled by remaining 3 digits .
So, hundred's place can be filled in 3 ways.
Similarly, to fill thousand's place, we have 2 digits remaining.
So,thousand's place can be filled in 2 ways.
So, required number of ways in which four digit even numbers can be formed from the given digits is $2\times 4\times 3\times 2=48$