A school has 100 members in three clubs…
Contributed by StemPy LabTry it on paper first. A hint, the key idea and the worked solution are here when you need them.
A school has members in three clubs. The robotics club has members, the debating club has and the chess club has . Of these, are in robotics and debating, are in debating and chess, are in robotics and chess, and are in all three clubs. Every number counts the members who belong to those clubs, and possibly to more. Find how many members belong to at least one club, to no club, to exactly two clubs and to exactly one club.
Need a reminder? The key ideaShow
Use , and . For three events add the three singles, take away the three pairs and add the triple. De Morgan: .
The worked solution
Have you tried it?
Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.
Step 1. At least one. .
Step 2. No club. .
Step 3. Exactly two. Each pair count includes the who are in all three. So the members in exactly two clubs are .
Step 4. Exactly one. Take the members in two clubs and in three clubs from the union: .
Answer: , , and members.
Check by regions. Robotics only is , debating only is and chess only is . These add up to .
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