A sports club has 3 departments, tennis, squash and badminton

Question

The question is:

A sports club has 3 departments, tennis, squash and badminton. We get given the following information.

• 90 people are members of the tennis department.

• 60 people are members of the squash department.

• 70 people are members of the badminton department.

• 25 people are members of both the tennis department and the squash department.

• 15 people are members of both the tennis department and the badminton department.

• 13 people are members of both the squash department and the badminton department.

• 4 people are members of all 3 departments.

How many different members does the sports club have?

I can not see the idea of how to solve it. Had there not been "repetition" then it would have been another talk. Anyone who can give hints? Am I correct if I think about of this formula? $C(n,r)=\frac{n!}{r!(n-r)!}$ Update: I might be wrong with the formula, since there is repetition. I believe it's $\frac{(n+r-1)!}{r!(n-1)!}$

Answer 1

Just set up a Venn diagram with $3$ circles, and place a variable in each of the $7$ bounded regions.

Note that you have $7$ pieces of given information, each of which yields a linear equation.

Thus, you'll have $7$ linear equations in $7$ unknowns.

Of course, if some piece of information just gives an immediate value for a given region, then you don't need a variable for that region or an equation for that piece of information.

Solve the system, and you'll easily get the required answer.

Answer 2

Start by drawing a Venn diagram like this:

where each circle represents one department. Now, start putting in numbers in the regions one by one as you can figure them out, letting each number represent the number of students in that exact region. Many of the numbers you have been given are actually the sum of the numbers in multiple regions, so you can't fill them in immediately. The only thing you can fill in initially is that center region where all circles overlap, which should have $4$ in it.

Next, there are $13$ people who do squash and badminton. That means that the two regions where the squash and badminton circles overlap (one region is where they also overlap the tennis circle, and one where they don't) should have $13$ in total. Since you already have $4$ in one region, that leaves $9$ for the part of the squash-badminton overlap which is outside the tennis circle.

Keep going like this, and eventually you will have filled each region with a number representing exactly how many are in each group.

Answer 3

Let $B$ denote the number of people who play badminton; let $S$ denote the number of people who play squash; let $T$ denote the number of people who play tennis. The number of people in the club is the number of people who play one of these sports, which by the Inclusion-Exclusion Principle, is $$|B \cup S \cup T| = |B| + |S| + |T| - |B \cap S| - |B \cap T| - |S \cap T| + |B \cap S \cap T|$$ Why should this be the case?

If we simply add up the members who play badminton, squash, or tennis, we will count each person who plays two sports twice, once for each sport they play. We only want to count them once, so we must subtract the number who play two sports from the total $|B| + |S| + |T|$. However, if they play three sports, we have both added them three times, once for each sport they play, and subtracted them three times, once for each pair of sports they play. Hence, the quantity $$|B| + |S| + |T| - |B \cap S| - |B \cap T| - |S \cap T|$$ does not count those who play three sports at all. Thus, we need to add the number of people who play all three sports to $|B| + |S| + |T| - |B \cap S| - |B \cap T| - |S \cap T|$ to account for them.

Answer 4

The idea of setting the Venn diagram (mentioned before) is very good, but I will modify it a bit:

Set it to $7$ regions like below ($T$ is for tennis, $S$ is for squash, $B$ is for badminton)

Assume the number of people in each region is $a_{1},a_{2},a_{3},a_{4},a_{5},a_{6},a_{7}$. I will state again the seven statements:

• $90$ people are members of the tennis department.

• $60$ people are members of the squash department.

• $70$ people are members of the badminton department.

• $25$ people are members of both the tennis department and the squash department.

• $15$ people are members of both the tennis department and the badminton department.

• $13$ people are members of both the squash department and the badminton department.

• $4$ people are members of all 3 departments.

We have this set of equations (respectively for each of the statements):

${\begin{cases}a_1+a_2+a_4+a_7=90\\a_2+a_3+a_5+a_7=60\\a_4+a_5+a_6+a_7=70\\a_2+a_7=25\\a_4+a_7=15\\a_5+a_7=13\\a_7=4\end{cases}}$

Replace $a_7=4$ for all the equations above, we will have:

${\begin{cases}a_1+a_2+a_4=86\\a_2+a_3+a_5=56\\a_4+a_5+a_6=66\\a_2=21\\a_4=11\\a_5=9\\a_7=4\end{cases}} \Rightarrow{\begin{cases}a_1+21+11=86\\21+a_3+9=56\\11+9+a_6=66\\a_2=21\\a_4=11\\a_5=9\\a_7=4\end{cases}} \Rightarrow{\begin{cases}a_1=54\\a_3=26\\a_6=46\\a_2=21\\a_4=11\\a_5=9\\a_7=4\end{cases}} $

That set of seven equations can be solved much quickly than you (and I) thought.

The number of different members in this sport club is:

$a_{1}+a_{2}+a_{3}+a_{4}+a_{5}+a_{6}+a_{7}=54+21+26+11+9+46+4=171$