30 people are interested in photography but not in music or swimming.
How many are interested in photography but not in music or swimming?
M: Set of people interested in music. (100 people)
P: Set of people interested in photography. (70 people)
S: Set of people interested in swimming. (40 people)
M∩P: Set of people interested in both music and photography. (40 people)
M∩S: Set of people interested in both music and swimming. (30 people)
P∩S: Set of people interested in both photography and swimming. (20 people)
M∩P∩S: Set of people interested in all three activities. (10 people)
The total number of people interested in photography, including those with overlapping interests, is:
n(P) = n(P) + n(M∩P) - n(P∩S) - n(P∩M∩S) + n(P∩M∩S)
n(P) = 70 + 40 - 20 - 10 + 10
n(P) = 80
We want to find the number of people who are interested in photography but not music or swimming. This is the difference between the total number interested in photography (n(P)) and those who are interested in either music or swimming or both (n(M) ∪ n(S) ∪ (M∩S)):
n(Photography Only) = n(P) - (n(M) ∪ n(S) ∪ (M∩S))
We know n(P) = 80. To calculate n(M) ∪ n(S) ∪ (M∩S), we can use the principle of inclusion-exclusion again:
n(M ∪ S ∪ (M∩S)) = n(M) + n(S) - n(M∩S)
n(M ∪ S ∪ (M∩S)) = 100 + 40 - 30
n(M ∪ S ∪ (M∩S)) = 110
Therefore, the number of people interested in photography only is:
n(Photography Only) = 80 - 110
n(Photography Only) = 30
Therefore, 30 people are interested in photography but not in music or swimming.