Answer to Question #145085 in Algebra for Kayla

Question

When a group of colleagues discussed where their annual retreat should take place , they found that of all the colleagues : 19 would not go to a park , 18 would not go to a beach 21 would not go to a cottage B would go to neither a park nora beach , 7 would go to neither a beach noca cottage 3 would go to neither park nor a cottage , 1 would not go to a park or a beach or a cottago , and 15 were willing to go to all three places . What is the total number of colleagues in the group ?

Answer 1

Let X be the subset of 19 people who would not go to the park.

Let Y be the subsets of 18 people who would not go to the beach.

Let Z be the subset of 21 people who would not go to the cottage.

Intersection XY, (X n Y) of the subsets X and Y contains B people, that is, neither a park nor a beach.

Intersection YZ, (Y n Z) of the subsets Y and Z contains 7 people, that is, neither a beach nor a cottage.

Intersection XZ, (X n Z) of the subsets X and Z contains 3 people, that is, neither a park nor a cottage.

Intersection XYZ complement, (X n Y n Z)' contains 1 person, that is, would not go to a park or a beach or a cottage.

Intersection XYZ, (X n Y n Z) contains 15 people, that is, willing to go to all three places.

Therefore: n(X u Y u Z) = n(X) + n(Y) + n(Z) - n(X n Y) - (Y n Z) - n(X n Z) + n(X n Y n Z)

n(X u Y u Z) = 19 + 18 + 21 - B -7 -3

+ 15

= 63 - B

Total number of colleagues in the group = n(X u Y u Z) + n (X u Y u Z)'

= 63 - B + 1

= 64 - B

Therefore, the total number of colleagues in the group = 64 - B.

Note: I think "B" seems to be a mistake. If it is, just go through the same process and input the correct value wherever you see "B".

If it isn't, "64 - B" is the final answer.