FUNDAMENTAL PRINCIPLE OF COUNTING PROBLEMS WITH SOLUTION

Problem 1 :

A person went to a restaurant for dinner. In the menu card, the person saw 10 Indian and 7 Chinese food items. In how many ways the person can select either an Indian or a Chinese food?

Solution :

Number of ways of selecting Chinese food items  =  7

Number of ways of selecting Indian food items  =  10

Here a person may choose any one food items, either an Indian or a Chinese food. So, we have to use "Addition" to find the total number of ways for selecting the food item.

Total number of selecting Indian or a Chinese food 

=  10 + 7  =  17 ways

Problem 2 :

There are 3 types of toy car and 2 types of toy train available in a shop. Find the number of ways a baby can buy a toy car and a toy train?

Solution :

According to the given question, a baby wants to buy both toy car and toy train. So we have to use the binary operation "Multiplication" to find the total number of ways.

Types of car available in the shop  =  3

Types of car available in the shop  =  2

Total number of ways of buying a car  =  3 (2)  =  6

Problem 3 :

How many two-digit numbers can be formed using 1,2,3,4,5 without repetition of digits?

Solution :

In order to form a two digit number, we have to select two numbers out of the given 5 numbers. 

___  ___

To fill up the first dash, we have 5 options.

To fill up the second dash, we have 4 options.

Number of two digit numbers formed using the above numbers  =  5 (4)  =  20.

Problem 4 :

Three persons enter in to a conference hall in which there are 10 seats. In how many ways they can take their seats?

Solution :

1st person may choose 1 seat out of 10 seats

2nd person may choose 1 seat out of 9 seats

3rd person may choose 1 seat out of 8 seats

Total number of ways of selecting seat  =  10 (9) (8)

  =  720 ways

Problem 5 :

In how many ways 5 persons can be seated in a row?

Solution :

5 persons may sit in 5 seats.

1st person may sit any one of the 5 seats 

2nd person may sit any one of the 4 seats and so on.

Hence the total number of ways  =  ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1

 =  120 ways

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