When a group of colleagues discussed where their annual retreat should take place, they found that all the colleagues: 25 would not go to a park, 27 would not go to a beach, 24 would not go to a cottage, 10 would go to neither a park nor a beach 7 would to neither a beach nor a cottage, 6 would go to neither a park nor a cottage, 4 would not go to a park or a beach or a cottage, and 6 were willing to go to all three places. What is the total number of colleagues in the group?

63 people

Step-by-step explanation:

We have the following:

Let P: the subset of those 25 who would not go to a park

Let B: the subset of those 27 who would not go to the beach

Let C: the subset of those 24 that would not go to a cottage

Thus:

The intersection PB of subsets P and B consists of 10 people (i.e. neither a park nor a beach)

The intersection BC of subsets B and C consists of 7 people (i.e. neither beach nor cottage)

The intersection PC of subsets P and C consists of 6 people (i.e. neither a park nor a cottage)

We are also told that the PBC intersection of subsets P, B, and C consists of 4 person (i.e. would not go to a park, beach, or cottage)

The complement of the union of the sets P, B and C for the whole group consists of 6 people (that is, willing to go to the three places)

The formula to apply is as follows:

# (PUBUC) = #A + #B + #C - #AnB - #BnC - #AnC + #AnBnC

Replacing:

 = 25 + 27 + 24 - 10 - 7 - 6 + 4

= 57 people

those who are willing to all who are 6 would be missing us, therefore:

57 + 6 = 63

The group consists of a total of 63 people.