Question

How many 3 digits even numbers can be made using the digits 1,2,3,4,5,6,7 if no digit is repeated.
From permutation and combination chapter

Answer 1

Numbers are : 1, 2, 3, 4, 5, 6, 7

We have to find the 3 digit even number.

Here .. even numbers are : 2, 4, 6.

A number which have 2, 4, 6 any one of them in last is considered as even number.

So, let the last number be 2.

___ , ___ , 2.

We have find 3 digits. But now I have last number is 2. Means we have to find first two.

From numbers : 1, 2, 3, 4, 5, 6, 7.

We take 2 so, left numbers are : 1, 3, 4, 5, 6, 7.

Now total number = 6.

We have to fill two more space or we have to find two more numbers.

So, r = 2

Now .. ${P}^{6}_{2}$

=> $\dfrac{6!}{(6\:-\:2)!}$

=> $\dfrac{6!}{4!}$

=> $\dfrac{6\:\times\:5\:\times\:4}{4}$

=> 30

30 is the three digit even number having unit place two.

Similarly, 30 is the three digit even number having unit place four and six.

Means ..

Total 3 digit even number → 30 + 30 + 30

→ 90

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90 is the 3 digits even number.

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