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?

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.

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DeanR DeanR · Answer 2 · 2018-07-01T04:57:34+02:00

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 [tex]P,B,C.[/tex]

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

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

[tex]\bar{P}\bar{B}\bar{C} = 1[/tex]

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.

[tex]8 = \bar{P}\bar{B}\bar{C} + \bar{P}\bar{B}C[/tex]

[tex]\bar{P}\bar{B}C=8-1=7[/tex]

3 would go to neither a beach nor a cottage,

[tex]P\bar{B}\bar{C}=3-1=2[/tex]

2 would go to neither a park nor a cottage,

[tex]\bar{P}B\bar{C} = 2-1=1[/tex]

17 would not go to a park

[tex]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}[/tex]

[tex]\bar{P}{B}{C}=8[/tex]

24 would not go to a beach

[tex]{P}\bar{B}{C}=24 - 1 - 7 - 2 = 14[/tex]

13 would not go to a cottage

[tex]{P}B\bar{C}=13 - 1 -2 - 1= 9[/tex]

8 were willing to go all three places.

[tex]{P}BC= 8[/tex]

Adding them all up,

[tex]8 + 9+ 14+ 8 + 1 + 2 + 7 + 1 = 50[/tex]