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?

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
 
We'll label the eight regions like , here meaning in P, in B, not in C.

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



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.

 

 

3 would go to neither a beach nor a cottage,

 

2 would go to neither a park nor a cottage, 

 

17 would not go to a park





24 would not go to a beach



13 would not go to a cottage
 


8 were willing to go all three places.



Adding them all up,

Okay so for every triangle has three angles that always add up to 180 degrees. and to tell the difference between whether they are secure, right or obtuse is that acute: has one angle that is less than 90 degrees. right triangle means there is at least one angle that is exactly 90 degrees. and an obtuse angle has an angle that is bigger than 180 degrees. i’ll do #13 as an example: 13. 40 and 50 so if every triangle has 180 degrees 40+50= 90 degrees then180-90= 90 degrees so the third angle is 90 degrees and because this triangle has a 90 degree angle it means it is a right triangle