Find the number of 4-digit numbers that can be formed using the digits
Question:

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

Solution:

Number of 4-digit numbers that can be formed using the digits 1, 2, 3, 4, 5 = Number of arrangements of 5 digits taken 4 at a time = 5P4 = 5! = 120

Now, these numbers also consist of numbers in which the last digit is an odd digit.

So, in order to find the number of even digits, we subtract the cases in which the unit’s digit have been fixed as an odd digit.

Fixing the unit’s digit as 1:

Number of arrangements possible $={ }^{4} P_{3}=4 !$

Fixing the unit’s digit as 3:

Number of arrangements possible $={ }^{4} P_{3}=4 !$

Fixing the unit’s digit as 5:

Number of arrangements possible $={ }^{4} P_{3}=4 !$

$\therefore$ Number of 4-digit even numbers that can be formed $=120-4 !-4 !-4 !=120-24-24-24=48$