Of 450 people interviewed, 135 contributed to Red House, 180 contributed to Blue House, and 180 contributed to Yellow House. 27 contributed to Red and Yellow. 54 contributed to Yellow and Blue. 45 contributed to Red and Blue. 63 of the people interviewed contributed to none of the houses. How many contributed to all three houses?

The number of people who contributed to all three houses is 18.

Step-by-step explanation:

Total number of people interviewed = 450

Number of people who contributed to red house = n (R) = 135

Number of people who contributed to Blue house = n (B) = 180

Number of people who contributed to Yellow house = n (Y) = 180

Also, according to question:

n (R ∩ Y) = 27, n (B ∩ Y) = 54 and n (B ∩ R) = 45

63 people contributed to NONE = 63

So, number of people who contribute at least to any 1 house

= Total people - 63 = 450 - 63 = 387

⇒ n (R ∪ B ∪ Y) = 387

Using : n (R ∪ B ∪ Y) = n (R) + n (B) + n (Y) - n (R ∩ Y) - n (B ∩ Y) - n (B ∩ R) + n (B ∩ R∩Y)

⇒ 387 = 135 + 180 + 180 - 27 - 54 - 45 + n (B ∩ R∩Y)

or, n (B ∩ R∩Y) = 387 - 495 + 126 = 18

or, n (B ∩ R∩Y) = 18

Hence, the number of people who contributed to all three houses is 18.