# Rule of Sum

The **rule of sum** is a basic counting approach in combinatorics. A basic statement of the rule is that if there are $n$ choices for one action and $m$ choices for another action, and the two actions cannot be done at the same time, then there are $n+m$ ways to choose one of these actions.

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## Basic Examples

The rule of sum only applies to choices that are mutually exclusive, meaning that only one of the choices can be picked. To determine when to use the rule of sum (as opposed to the rule of product), try to rephrase the question. If the question can be rephrased with the word "or," this usually indicates that the rule of sum applies.

Mary is wearing her lucky shirt today, and she has to choose among 3 red skirts and 4 blue skirts to wear with the shirt. How many different outfit choices of one skirt does she have for the day?

Since Mary can wear one of the 3 red skirts

orone of the 4 blue skirts, the choices are mutually exclusive and the rule of sum applies. This gives a total of $3 + 4 = 7$ different outfit choices. $_\square$

Ravi goes to a pet shop and finds that the pet shop has $3$ reptiles, $4$ birds, $5$ rabbits, and $6$ fish. If Ravi can only pick one animal as a pet, how many choices does Ravi have for a pet?

Since Ravi can choose a reptile,

ora bird,ora rabbit,ora fish, the rule of sum applies. Then

- there are $3$ ways to select a reptile;
- there are $4$ ways to select a bird;
- there are $5$ ways to select a rabbit;
- there are $6$ ways to select a fish.
By the rule of sum, there are $3+4+5+6=18$ ways to select a pet. $_\square$

Chris is playing a card game and the cards in his hand include three 5's, two Jacks, two Aces, one 9, and one King. If he has to choose one card to play in the next round, how many choices does he have for a card to play?

Since Chris can choose to play a $5,$

ora Jack,oran ace,ora 9,ora King, the rule of sum applies. Then

- there are $3$ ways to select a $5;$
- there are $2$ ways to select a Jack;
- there are $2$ ways to select an Ace;
- there is $1$ way to select a $9;$
- there is $1$ way to select a King.
By the rule of sum, there are $3+2+2+1+1=9$ ways to select a card to play in the next round. $_\square$

At the dealership in Isaac's town, there are 10 red trucks, 5 blue trucks, 3 red cars, and 2 blue cars for sale. If Isaac is going to buy exactly one red vehicle, how many choices does he have?

The vehicles that are red are the 10 red trucks and 3 red cars. Hence, there are $10+3= 13$ red vehicles in total. $_\square$

Given a complete deck of cards, how many of the cards are either black face cards, or red and even?

Since there are 3 face cards (Jack, Queen, and King), and 5 even cards (2, 4, 6, 8, 10), we have the following that meets our criteria:

- 3 face cards in spades
- 3 face cards in clubs
- 5 evens in hearts
- 5 evens in diamonds.
Thus, by the rule of sum, there are $3 + 3 + 5 + 5 = 16$ choices that would work. $_\square$

## Problem Solving

How many non-negative integer solutions are there to the following:

$-5 < x < 5\quad \text{ or }\quad 12 < x < 100 ?$

Looking at each inequality separately, we see that there are $4 - (-4) + 1 = 9$ integer solutions to $-5 < x < 5$, but $-4, -3, -2,$ and $-1$ are all negative, so there are only $9-4=5$.

There are $99-13+1 = 87$ integer solutions to $12 < x < 100$, and they are all positive.

Thus, by the rule of sum, we have $5 + 87 = 92$ possible answers. $_\square$