Answer :

**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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