Question

(a) Explain when to use addition and when to use multiplication in combinatorics. Give examples for both.

(b) Give one example of a problem that requires both addition and multiplication to count the number of ways to do something.

Answer 1

To know better about this, Let’s understand with the help of two questions:

1. Adi has 3 different types of shoes and 2 different types of sandals. Whenever she goes out, she likes to wear either a shoe or a sandal. In how many ways can she decide what to wear?
2. Akki has 3 different types of shirts and 2 different types of trousers. Whenever he goes out, he likes to wear a shirt and a trouser. In how many ways can he decide what to wear?

Look at these two situations. In both these cases, the numbers are the same. The only difference is that Adi will choose either shoes OR sandals to wear and in the second case Akki is going to choose both a shirt AND a trouser.

Let us focus on Adi’s situation first, and then we will come to Akki.

Assuming that she has red, green, and black shoes, and she has brown and blue sandals, let us list down all the possible options that she has.

1. She can wear Red Shoes OR
2. She can wear Green shoes OR
3. She can wear Black Shoes OR
4. She can wear Brown Sandals OR
5. She can wear Blue Sandals

Notice how we have used the word OR after each and every case. The OR here emphasizes on the fact that Adi does not have the option to wear two different kinds of things at the same time! She needs to choose only 1 of these.

Thus, the answer, in this case, will be 5.

Whenever we have a situation in which two events cannot occur simultaneously, we simply add all the cases. Thus, in this case, we will say:

• Total possible cases for Adi = She will wear shoes OR she will wear Sandals
  • Number of ways she can wear shoes = 3
    Number of ways she can wear Sandals = 2
  •Therefore, total possible cases = 3 OR 2 = 3 + 2 = 5

Now let us look at Akki’s case.

Assuming that he has red, green, and a black shirt, and he has brown and blue trousers, let us list down all the possible options that he has.

1. He can wear Red Shirt AND Brown Trousers
2. He can wear Red Shirt AND Blue Trousers
3. He can wear Green shirt AND Brown Trousers
4. He can wear Green shirt AND Blue Trousers
5. He can wear Black shirt AND Brown Trousers
6. He can wear Black shirt AND Blue Trousers

Notice how I have used the word AND after each and every case.
The AND here emphasizes on the fact that Akki has to wear both shirt AND trousers at the same time.

From the above cases, we can see that he has 6 different options for wearing a shirt and a trouser.

Whenever we have a situation in which two events can happen simultaneously, we simply “multiply” all the cases.
Thus, in this case, we can say:

• Total possible cases for Akki = He will wear a Shirt AND he will wear Trousers
  • Number of ways he can wear a shirt = 3 ( Red OR Green OR Black) [ Keyword OR: hence addition]
    Number of ways he can wear trousers = 2 (Blue OR Brown)
  
  •Therefore, total possible cases = 3 AND 2 = 3 x 2 = 6

So I will conclude my study with the following two points:.

1. Whenever we come across a situation involving 2 or more events, and occurrence of one event does not affect the occurrence of the other event, i.e., both of the events cannot occur simultaneously, then, in that case, we will simply add up all the events. Also Look for the word OR in the question while figuring out what you need to find out and if OR is present then that means you need to add up the events.

2. Whenever we come across a situation involving 2 or more events and each event can happen simultaneously, i.e., event 1, event 2, event 3 and so on, all can happen simultaneously.Then, in that case, we will simply MULTIPLY up all the events. Also Look for the word AND in the question while figuring out what you need to find out, and if AND is present, then that means you need to multiply the events.