HOW MANY 3 DIGIT NUMBERS CAN BE FORMED WITH GIVEN DIGITS

(i) no restriction.

Thousands place :

Other than 0, we have 9 options to fill up the thousands place.

Hundreds place :

Including 0, we have 10 options.

Tens place :

Including 0, we have 10 options.

Unit place :

Including 0, we have 10 options.

Hence total numbers lie between 999 and 10000 is 

9 ⋅ 10 ⋅ 10 ⋅ 10  =  9000

(ii) no digit is repeated.

Thousands place :

Other than 0, we have 9 options to fill up the thousands place.

Hundreds place :

Here we may accept 0, but we should not use the number that we have already used in the thousands place. So, we have 9 options.

Tens place :

We have 8 options.

Unit place :

We have 7 options.

So, the total numbers are 

9 ⋅ 9 ⋅ 8 ⋅ 7  =  4536

(iii) at least one of the digits is repeated

At least one of digits is repeated means, we may have one repeated digit, two repeated digits or more.

In order to find the answer for this question, we have to subtract no digits repeated from total numbers.

Hence the answer is 9000 -  4536  =  4464.

(i) repetition of digits are not allowed?

Case 1 :

If we use 0 in the unit place, we will have only one option.

In tens place, we will have 5 options.

In hundreds place, we will have 4 options.

Case 2 :

If we use 5 in the unit place, we will have only one option.

In hundreds place other than 0 and 5, we will have 4 options.

In tens place, we will have 4 options including 0.

Hence total numbers formed  =  20 + 16  =  36 numbers

(ii) repetition of digits are allowed?

Unit place :

Since the required numbers are divisible by 5, it ends with 0 or 5. So, we have 2 options.

Hundreds place :

Other than 0, we will have 5 options.

Tens place :

Sine the repetition is allowed , we have 6 options.

Hence total numbers to be formed  =  5 ⋅ 6 ⋅ 2

  =  60 numbers

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