Combinations Question Help (Please provide explanation)

Question

A drama club is putting on two plays. There are 11 actors in the Oz play and 7 in the

Grease play.

1. a) If 3 actors are in both plays, how many actors are there in all?

b) Use a Venn diagram to calculate how many students are only in one of the two plays.

Please explain for each scenario, I don't understand the answers to these, what is the reason??

Answer 1

There are 3 actors in both plays. If there are 11 actors in Oz, then there are 11-3=8 actors who are only in Oz, and not in Grease. If there are 7 actors in Grease, then there are 7-3=4 actors who are only in Grease, and not in Grease.

8 actors only in Oz + 4 actors only in Grease + 3 actors in both Oz and Grease = 15 actors in total

The Venn diagram would have 8 in Oz alone, 4 in Grease alone, and 3 in the overlap.

Answer 2

Number in Oz play:

O + B = 11

Number in Grease play:

G + B = 7

Number in both plays:

B = 3

b) only in one play:

O = 8

G = 4

O + G = 12

a) How many actors in all?

O + B + G = 8 + 3 + 4 = 15