# Usage of Sum and Product Rule

In case exactly one of the specified events occurs, we apply the sum rule. In case each of the specified event occurs in the given order, we apply the product rule.

**Example:**A house has 4 doors and 10 windows.

- In how many ways can a burglar enter the house, if he may enter through a door of a window?
- In how many ways can the burglar rob the house by entering through a window and exiting through a door?

**Solution:**

Two events are involved: Selecting a window and selecting a door. The first event can occur in 10 different ways and the second event can occur in 4 different ways.

- Since the burglar uses either a door or a window to enter the house,
*exactly*one of the two events occur. By the sum rule, the burglar has 4 + 10 = 14 choices to enter the house. - Since the burglar enters through a window and exits through a door, both the events must occur. Therefore, by the product rule, the burglar has 10 Ã— 4 = 40 possible ways to rob the house.

**Example:**Sheela - Vishan restaurant offers a choice of 4 different kinds of pizzas and 5 different kinds of burgers.

- If Rakshit wants to have either a pizza or a burger, how many different choices does he have?
- If Rakshit wants to have both a pizza and a burger, how many different choices does he have?

**Solution:**

Two events are involved: Selecting a pizza and selecting a burger. The first event can occur in 4 different ways and the second events can occur in 5 different ways.

- Since Rakshit wants to have either a pizza or a burger, exactly one of the two events occur. By the sum rule, Rakshit has 4 + 5 = 9 possible ways to choose a pizza or a burger.
- Since Rakshit wants to have both a pizza and a burger, both the events must occur. By the product rule, Rakshit has 4 x 5 = 20 possible ways to choose a pizza and a burger.

**Example:**A class contains 15 boys and 10 girls.

- In how many ways can a student be chosen from the class, to represent the class in the games committee?
- In how many ways can a boy and a girl be chosen from the class to represent the class the student
^{'}s council?

**Solution:**

Two events are involved: Selecting a boy and selecting a girl. The first event can occur in 15 different ways and the second event can occur in 10 different ways.

- Since only one student is to be selected to represent the class in the games committee, exactly one of the two events must occur. By the sum rule, there are 15 + 10 = 25 ways to choose a student to represent the class in the games committee.
- Since a boy and a girl are to be chosen from the class to represent the students council, both the events must occur. Therefore, by the product rule, there are 15 x 10 = 150 ways to choose a boy and a girl to represent the student
^{'}s council.

**Example:**In a school there are 30 students in Commerce section and 50 students in Science section.

- In how many ways can a student be selected to represent the school on Mathemania (Mathematics Quiz contest)?
- In how many ways can a student from the Commerce section and a student from the Science section be selected to form a team of two to represent the students on Mathemania?

**Solution:**

Two events are involved: Selecting a student from the Commerce section and selecting a student from Science section. The first can occur 30 ways and the second event can occur in 50 ways.

- Since only one student is to be selected, exactly one of the two events must occur. By the sum rule, there are 30 + 50 = 80 ways to choose a student to represent the school on Mathemania.
- Since a student from the Commerce section and a student from the Science section are to be selected to form a team of two, both the events must occur. By the product rule there are 30 ? 50 = 1500 ways to choose a team of two students to represent the school on Mathemania