# Finding the number of people that like each ice cream

So 300 individuals were asked in a survey what kind of scoops of ice cream they would like with a minimum of 1 flavor and a maximum of 2 flavors.

(chocolate) A = 122 people

(strawberry) B = 150 people

(vanilla) C = 62 people

(chocolate and strawberry) A ∩ B = 38 people

(chocolate and vanilla) A ∩ C = 20 people

(strawberry and vanilla) B ∩ C = 28 people

(Do not want ice cream) 36 people

I want to find how many people only want one scoop of each flavor.

i.e. How many people only want chocolate? How many people only want Vanilla? How many people only want strawberry?

So far, I am as far as figuring out that only 264 people wanted ice cream. And have drawn a venn diagram, and am now stuck.

• The numbers don't add up. Please double check that all numbers are accurate and/or that all conditions stated are accurate. In particular "donot want icecream = 36" and "minimum 1 flavor and a maximum of 2 flavors" – JMoravitz Apr 29 '15 at 12:08

"Minimum of one flavor and maximum of two flavors" implies $A\cap B\cap C = \emptyset$ since noone was allowed to say they like all three flavors.

We apply the principle of inclusion-exclusion here:

The principle of inclusion exclusion: $|A\cup B| = |A| + |B| - |A\cap B|$

Furthermore: $|A\cup B\cup C| = |A|+|B|+|C|-|A\cap B|-|A\cap C|-|B\cap C| + |A\cap B\cap C|$

And more generally: $|A_1\cup A_2\cup \dots \cup A_n| = \sum\limits_{i=1}^n(-1)^{i+1}\sum\limits_{I\subset [n]~:~|I|=i} |\bigcap\limits_{k\in I}A_k|$

So, let us try to figure out some of the missing information.

We know that $|A\cup B| = |A|+|B|-|A\cap B| = 122+150-38 = 234$, that $|A\cup C| = |A|+|C|-|A\cap C| = 122 + 62-20 = 164$ and that $|B\cup C| = 150 + 62 - 28 = 184$

We also know that $|A\cup B\cup C| = |A|+|B|+|C|-|A\cap B|-|A\cap C| - |B\cap C| + |A\cap B\cap C| = 122 + 150 + 62 - 38 - 20 - 28+0=248$

(at this point I notice that there is an error in the problem statement, as this does not agree with one of your statements, either the number of people total, that noone wants all three flavors, or the number of people not wanting any icecream. I will continue anyways, as the method is what is more important)

We are curious to find $|A\setminus(B\cup C)|$ (i.e. the people who want only chocolate and no other flavor). Notice that $(A\setminus(B\cup C))\cup (B\cup C) = A\cup B\cup C$ and that $(A\setminus (B\cup C))$ is disjoint from $(B\cup C)$. So, by inclusion-exclusion, $|A\setminus(B\cup C)|= |A\cup B\cup C| - |B\cup C| = 248 - 184 = 64$

Similarly, $(B\setminus(A\cup C))\cup (A\cup C) = B\cup A\cup C$ and $|B\setminus (A\cup C)| = |A\cup B\cup C|-|A\cup C| = 248 - 164 = 84$

Do so similarly for calculating $|C\setminus (A\cup B)|$.

Note: As there is an error someplace in the problem statement, these numbers might not be entirely accurate. The general procedure still holds however. I expect that the error was in writing the statement "minimum of 1 flavor, maximum of 2 flavors", in which case my value for $|A\cap B\cap C|$ was incorrect, throwing my calculation for $|A\cup B\cup C|$ off as well. Assuming the statement was correct for "there are 300 people surveyed" and "there are 36 people who don't want any icecream", then you would arrive at $|A\cup B\cup C| = 300-36$ (which in turn implies that there some people who want all three flavors).

Continue again by calculating each of $|A\cup B|,|A\cup C|,|B\cup C|$ and then by calculating $|A\setminus (B\cup C)| = |A\cup B\cup C|-|B\cup C|$.

Try a contingency table : http://en.wikipedia.org/wiki/Contingency_table