Question

When a group of colleagues discussed where their annual retreat should take​ place, they found that of all the​ colleagues: 17 would not go to a​ park, 24 would not go to a​ beach, 13 would not go to a​ cottage, 8 would go to neither a park nor a​ beach, 3 would go to neither a beach nor a​ cottage, 2 would go to neither a park nor a​ cottage, 1 would not go to a park or a beach or a​ cottage, and 8 were willing to go to all three places. What is the total number of colleagues in the​ group?

Answer 1

We're assuming e.g. the two that won't go to P (park) or C (cottage) includes the one that won't go anywhere.

Let's call the three rings of the Venn diagram
`P,B,C.`

We'll label the eight regions like
`PB\bar{C}`, here meaning in P, in B, not in C.


1 would not go to a park or a beach or a​ cottage


`\bar{P}\bar{B}\bar{C} = 1`

8 would go to neither a park nor a​ beach. We have to take away the one guy who won't go anywhere to find out how many go just to college.


`8 = \bar{P}\bar{B}\bar{C} + \bar{P}\bar{B}C`


`\bar{P}\bar{B}C=8-1=7`

3 would go to neither a beach nor a​ cottage,


`P\bar{B}\bar{C}=3-1=2`

2 would go to neither a park nor a​ cottage,


`\bar{P}B\bar{C} = 2-1=1`

17 would not go to a​ park


`17 = \bar{P}\bar{B}\bar{C} + \bar{P}\bar{B}{C} + \bar{P}{B}\bar{C} + \bar{P}{B}{C}= 1 + 7 + 1 + \bar{P}{B}{C}`


`\bar{P}{B}{C}=8`

24 would not go to a​ beach


`{P}\bar{B}{C}=24 - 1 - 7 - 2 = 14`

13 would not go to a​ cottage


`{P}B\bar{C}=13 - 1 -2 - 1= 9`

8 were willing to go all three places.


`{P}BC= 8`

Adding them all up,


`8 + 9+ 14+ 8 + 1 + 2 + 7 + 1 = 50`

Answer 2

To find the answer look for opinions where 1 person can't have multiple.

Add them all up and you'll get your answer, there are 33 total colleagues in the group.