# How many 3-digit even numbers can be made using the digits 1, 2, 3, 4, 5, 6, 7 if no digits is repeated?

Given: Digits which can be used to make numbers are 1, 2, 3, 4, 5, 6 and 7

The number of these digits are 7

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

Even numbers are those numbers whose unit’s place is even

fix the position of 1 even number at unit’s place at one time

Even numbers are: 2, 4 and 6

Case 1:

Fix position of 2 at unit’s place Remaining numbers = 6

Arrange these 6 numbers at remaining 2 places

Formula used:

Number of arrangements of n things taken r at a time = P(n, r) The total numbers ending with 2 are =

= the number of arrangements of 6 things taken 2 at a time

= P(6, 2)   = 6 × 5

= 30

Case 2:

Fix position of 4 at unit’s place Remaining numbers = 6

Now, arrange these 6 numbers in remaining 2 places

Formula used:

Number of arrangements of n things taken r at a time = P(n, r) The total numbers ending with 2 are =

= the number of arrangements of 6 things taken 2 at a time

= P(6, 2)   = 6 × 5

= 30

Similarly,

When you fix position of 6 at unit’s place, 30 more numbers will be formed.

Hence, total number of three-digit even numbers with no digit repeated are, 30 + 30 + 30 = 90

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