Question

- Prove that there are exactly 4 ! numbers between 1000
and 10000 that contain the digits 1, 3, 5 and 7.

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Answer 1

Given:

Range of the numbers: 1000 to 10000

To find:

We need to find the numbers which contains the digits 1, 3, 5 and 7.

Solution:

According to the question, range of the numbers is 1000 to 10000.

It means that it is a 4 digit number.

Let the four digit number is _ _ _ _.

Given that, it can contain only 1, 3, 5 and 7.

So, first blank can be filled in 4 ways.

Now we have 3 numbers because in first blank already a number is chosen.

Similarity, the second blank is filled with 3 ways.

Third blank is filled with 2 ways.

Fourth blank is be filled with 1 way.

So, total number of ways $=4\times3\times2\times1$

The above value is nothing but $4!$.

Therefore, the number of ways is equal to $4!$.