1. How many Athletes?

This High School has the following numbers of athletes on four of its boys' sports teams:
41 football players; 15 basketball players; 21 baseball players; 32 wrestlers
No wrestler is also a basketball player. 9 boys play both football and baseball. 5 boys wrestle and play baseball.
12 boys wrestle and play football.
6 boys play basketball and football.
7 boys play basketball and baseball.
3 boys are on all of the teams except basketball. 4 boys are on all of the teams except wrestling.
How many boys are involved in these four sports?

2. Re: How many Athletes?

Are you familiar with a concept known as the Inclusion-Exclusion principle? Let A, B, C, and D be the sets of all football players, basketball players, baseball players, and wrestlers respectively. You can think of them like lists with the player's names written on them. When I write $\displaystyle |A|$, I mean the number of names on that list. When I write $\displaystyle A\cap B$, that means a new list of names where we only write down the names that appear on both the football list and the basketball list. When I write $\displaystyle |A\cap B|$, I mean the number of names on that list. The Inclusion-Exclusion principle states that the number of boys involved in these four sports is:

\displaystyle \begin{align*}& (|A| + |B| + |C| + |D|)\\ - & (|A\cap B| + |A \cap C| + |A \cap D| + |B \cap C| + |B \cap D| + |C\cap D|) \\ + & (|A\cap B \cap C| + |A \cap B \cap D| + |A \cap C \cap D| + |B \cap C \cap D|) \\ - & (|A\cap B \cap C \cap D|)\end{align*}

3. Re: How many Athletes?

Mr. SlipEternal. I am not familiar with this kind of principle. May I ask for further assistance? I know I am suppose to put the numbers of each players in the alphabets but what's the upside down U symbol stand for?

4. Re: How many Athletes?

Originally Posted by SuperMaruKid
Mr. SlipEternal. I am not familiar with this kind of principle. May I ask for further assistance? I know I am suppose to put the numbers of each players in the alphabets but what's the upside down U symbol stand for?
Can you at least tell us what each of these numbers are?
$\displaystyle \\ (|A\cap B|,~ |A \cap C|,~ |A \cap D|,~ |B \cap C|,~ |B \cap D|,~ |C\cap D|) \\ (|A\cap B \cap C|,~|A \cap B \cap D|,~|A \cap C \cap D|,~|B \cap C \cap D|) \\ (|A\cap B \cap C \cap D|)$

That is what are each value of those eleven numbers?

A : 41
b : 15
c : 21
d : 32

?

6. Re: How many Athletes?

The upside down U symbol is just like I explained. It means take the lists and find only names that are on each list that are connected with upside down U's. So, $\displaystyle A\cap B$ means only names of kids on both the football list and the basketball list. According to your numbers, there are six kids who play both football and basketball, so there are six names on that list. $\displaystyle A\cap B \cap C$ is the list of all kids who play football, basketball, and baseball. Your numbers say there are four boys who are on all teams except wrestling (all teams except wrestling means football, basketball, and baseball).

The first line is telling you to add up the number of names of kids on each sport's individual roster. But, if you do this, you are counting some kids multiple times (once for each roster they are on). Suppose a kid is on two rosters. Then you counted him twice. So, we subtract off all names that are on two lists. So, if a kid was on two lists, then you counted him twice initially, but subtracted him once when you subtracted the number of kids on two lists.

What happens if a kid was on three lists? Then you counted him three times initially. But if he is on three lists (say A, B, and C, for example), then he will be on the list of kids on both A and B: $\displaystyle A\cap B$. He will be on the list of kids on both lists A and C $\displaystyle A\cap C$, and he will be on the list of kids on both lists B and C $\displaystyle B \cap C$. So, when you subtract all kids on two lists, you subtract the kid three times. So now you haven't counted the kid at all. So, add all of the kids on three lists. Now you counted him once.

What happens if a kid was on four lists? (This is the last case, I promise). Now, you initially count the kid four times. But, he is on all six of the lists for kids that are on two lists. So, when you subtract him from those lists, you have counted him -2 times! So, when you add the kids that are on three lists, he is on all of the lists, so he is in all four of the lists of kids of three lists. So, you count him four times (4 - 6 + 4) = 2. Now you have overcounted him again by one. So, subtract the number of kids on four lists, and you counted him exactly once.

So, from your data, we have
\displaystyle \begin{align*} & (41 + 15 + 21 + 32) \\ - & (6 + 9 + 12 + 7 + 0 + 5) \\ + & (4 + 0 + 3 + 0) \\ - & (0) \end{align*}

7. Re: How many Athletes?

Another way of doing it: a Venn diagram. Draw 4 overlapping circles. Label them "W" (for "wrestling"), "Bk" (for "basketball"), "Bs" (for "baseball"), and "F" (for "football").

Since all four circles overlap, there will be a region that is in all 4 sets (That would be "$\displaystyle W\cup Bk\cup Bs\cup F$", the intersection of all four sets). Since "No wrestler is also a basketball player" there is no one in that region- leave it blank or write "0" in it.

"3 boys are on all of the teams except basketball." So write "3" in the region of overlap of "W", "Bs", and "F".

"4 boys are on all of the teams except wrestling." So write "4" in the region of overlap of "Bs", "Bk", and "F".

" 5 boys wrestle and play baseball." But that includes the 3 boys that are "on all teams except basketball" so there are 5- 3= 2 boys in the overlap of "W" and "Bs" only. Write "2" in the overlap of those two sets only.

"12 boys wrestle and play football." But that includes the 3 boys that are "on all teams except basketball" so there are 12- 3= 9 boys in the overlap of "W" and "F". Write "9" in the overlap of those two sets only.

Continue like that with the rest of the information, writing in each area the number of boys satisfying the conditions, minus the number already calculated, that satisfy the conditions and another. The objective is to have each number give the number of boys in that area and NO other. That way you can just add all of them to find the total number of athletes.

8. Re: How many Athletes?

Can you share some tutorials for my kid he is very much interested in this stuff and i wanted to learn it by himself.
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