ayakik wrote:
Hello
KarishmaB avigutman I have gone through the thread twice and I'm still having trouble understanding this concept.
I got the correct answer through the following process:
P+H+W = 77 - 59 = 18
18 - 2*6 = 6 (since 6 people are in 2 groups)
So remaining 6 must be in 3 groups
Is this reasoning correct? I'm a little confused about how to understand the general formula, Total=P+H+W -{Sum of Exactly 2 groups members}-2*PnHnW + Neither.
I always find both of your explanations so intuitive and so much easier to understand than general formulas so it would be really great if I could get your take on this. Thank you.
Just about but not quite correct.
You are thinking in terms of people and instances - which is great! There are total 59 people and 77 instances (number of choices made by these 59 people. Some people choose only one, some choose two and some choose 3 so we have more total choices than number of people)
Since everyone chooses at least one, 77 - 59 = 18 instances are left. These are of those people who choose two or three clubs. Since we have already removed one club for each person, now the people who chose 2 clubs are counted in 18 once. So we are left with 18 - 6 = 12 instances.
These 12 instances represent the people who chose 3 clubs. They will have two representations each in these 12 instances.
So number of people who chose 3 clubs = 12/2 = 6.
Alternatively, use the Venn diagram.
Attachment:
Screenshot 2022-11-14 at 10.19.27 AM.png [ 60.64 KiB | Viewed 7745 times ]
Say each letter represents the region of a single colour only. So (d + f+ e) represents people who chose exactly two clubs.
59 = a + b + c + d + e + f + k (each person lies somewhere in the region)
22 + 27 + 28 = (a + d + f + k) + (b + d + k + e) + (c + f + k + e)
77 = a + b + c + 2(d+f+e) + 3k
77 = 59 + (d + e + f) + 2k
77 = 59 + 6 + 2k
k = 6