College Mathematicsinclusion-exclusion principle

Counting sounds simple, but it can be tricky because categories can “overlap”.If we just add all up the categories, we'll count some things more than once.

Example 1:You're planning a party for 100 guests.75 said they like chicken.50 said they like mushrooms.7 said they like neither chicken nor mushrooms.

Deduce the number of guests who like mushrooms but don't like chicken.

A two-way table is helpful here. This situation has four non-overlapping categories,counted within the boxes of the table.Sub-totals and the grand total are on the far right or below the bottom row.

First, carefully fill in the numbers you know (shown in boldface).Continue to fill in numbers that can be calculated until the table is complete.

like chicken don't like chicken

like mushrooms 32 18 50

don't like mushrooms 43 7 50

75 25 100

Then, look in the table to find the information you want:Exactly 18 guests said they like mushrooms but don't like chicken.

A Venn diagram can also be useful for visualizing the situation:

Notice that sub-totals are usually not shown in a Venn diagram.

One more way of solving the problem is to use the inclusion-exclusion principle:

#(A or B) = #(A) + #(B) – #(A and B)

The number of items in A or B is the number of items in A plus the number in B minus the number in A and B (“the overlap”), since they've been counted twice.

100 guests

like chicken like mushrooms

743

1832

Example 2:Each of 350 people were classified by height (short/tall), hair (blonde/brunette), body (thick or thin). Here are some of the results:

There are 10 short, thick blondes.There are 75 tall blondes.There are 50 tall, thick brunettes.There are 160 tall people.There are 105 thin blondes.There are 170 blondes.There are 165 thin people.

With three different classifications, a three-way table is useful for analyzing this situation. This involves eight different non-overlapping categories, shown inthe boxes below.

short people tall people

thick thin thick thinblonde 10 85 95 blonde 55 20 75brunette 70 25 95 brunette 50 35 85

80 110 190 105 55 160

short and tall combined

thick thinblonde 65 105 170brunette 120 60 180

185 165 350

All of this is considered to be a single table. In particular, the "short and tall" combined data at the bottom only contains sub-totals and the grand total.

A three-set Venn diagram may also be helpful in this situation:

The inclusion-exclusion principle can be extended to involve three sets:

#(A or B or C) = #(A) + #(B) + #(C)– #(A and B) – #(A and C) – #(B and C) + #(A and B and C)

short people thick people

35

350 people

25

10

blondes

85

70

55

20

50