FUNDAMENTAL COUNTING PRINCIPLE PRACTICE QUESTIONS

Question 1 :

How many three-digit numbers are there with 3 in the unit place? (i) with repetition (ii) without repetition. 

Solution :

The following numbers are used to form any number.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9

By using the above numbers, we have to form a three digit number. In which unit place must contain the number "3".

(i)  with repetition 

  Unit place      ---->    must be 3  ---> 1 option

Ten's place :   

Including 3, we have 10 options

Hundred's place :    

 It should not contain 0. So, we have 9 options

Total numbers can be formed  =  9 ⋅ 10 ⋅  1  =  90

(ii) without repetition. 

   _____     _____     _____

  Unit place      ---->    must be 3  ---> 1 option

Hundred's place :

By excluding 3 and 0, we have 8 options.

Ten's place :

Excluding 3  and the number filled in the hundreds place, we have 8 options.

Total numbers can be formed  =  8 ⋅ 8 ⋅  1  =  64

Question 2 :

How many numbers are there between 100 and 500 with the digits 0, 1, 2, 3, 4, 5 ? if (i) repetition of digits allowed (ii) the repetition of digits is not allowed.

Solution : 

We must form a three digit number.

(i) repetition of digits allowed 

Unit place --> It may contain any of 6 numbers --> we have 6 options

Tens place --> It may contain any of 6 numbers --> we have 6 options

Hundreds place --> excluding 0 and 5 --> we have 4 options

Note : Any three digit number starts with 0. It should not 5, because we have to form a three digit number less than 500.

Total numbers can be formed  =  4 ⋅ 6 ⋅  6  =  144

(ii) the repetition of digits is not allowed.

Hundreds place :

Excluding 0 and 5 --> we have 4 options

Tens place :

Since repetition in not allowed including 0, we have 5 options.

Unit place :

Excluding the number filled in the tens and hundreds place, we have 4 options.

Total numbers can be formed  =  4 ⋅ 5 ⋅  4  =  80

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