# Find the numbers 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?

Given, digits which can be used to make numbers are 1, 2, 3, 4 and 5

The number of these digits are 5

To find: total number of 4-digit numbers with no digit repeated

Formula used:

Number of arrangements of n things taken r at a time = P(n, r)

The total number of ways

= the number of arrangements of 5 things taken 4 at a time

= P(5, 4)

= 120

Hence, total number of 4-digit numbers using digits 1 to 5 with no digit repeated are 120

Now, for 4-digit even number from digits 1, 2, 3, 4 and 5:

Let 4-digit even number be

Fix the position of unit’s place i.e. t as an even number for which we have 2 choices (2 or 4)

Now, for position x we have remaining 4 choices

Similarly, for position y and z we have 3 and 2 choices respectively

Total number of even numbers are

= multiplication of choices of x y z t

= 4 × 3 × 2 × 2

= 48

Hence, total number of 4-digit numbers using digits 1 to 5 with no digit repeated are 48

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