The management of a hotel conducted a survey. It found that of the 2560 guests who were surveyed, 1785 tip the wait staff. 1219 tip the luggage handlers. 831 tip the maids. 275 tip the maids and the luggage handlers. 700 tip the wait staff and the maids. 755 tip the wait staff and the luggage handlers. 245 tip all three services. 210 do not tip these services. How many of the surveyed guests tip a. exactly two of the three services
To find out how many of the surveyed guests tipped exactly two of the three services, we need to subtract the number of guests who tipped all three services from the sum of the guests who tipped exactly one service.
First, let's calculate the number of guests who tipped exactly one service:
Number of guests who tipped the wait staff only = 1785 - 700 - 755 - 245 = 85
Number of guests who tipped the luggage handlers only = 1219 - 700 - 755 - 245 = 19
Number of guests who tipped the maids only = 831 - 700 - 245 - 275 = -389 (since negative doesn't make sense, we assume there are 0 guests)
Now, let's calculate the number of guests who tipped exactly two services:
Number of guests who tipped the wait staff and the luggage handlers only = 755 - 245 = 510
Number of guests who tipped the wait staff and the maids only = 700 - 245 = 455
Number of guests who tipped the luggage handlers and the maids only = 275 - 245 = 30
Now, we can calculate the total number of guests who tipped exactly two of the three services:
85 + 19 + 0 + 510 + 455 + 30 = 1099
Therefore, 1099 of the surveyed guests tipped exactly two of the three services.
To find the number of guests who tip exactly two of the three services, we can use the principle of inclusion-exclusion.
Let's define:
A = guests who tip the wait staff
B = guests who tip the luggage handlers
C = guests who tip the maids
We know:
n(A) = 1785 (number of guests who tip the wait staff)
n(B) = 1219 (number of guests who tip the luggage handlers)
n(C) = 831 (number of guests who tip the maids)
n(A ∩ B) = 755 (number of guests who tip both the wait staff and the luggage handlers)
n(A ∩ C) = 700 (number of guests who tip both the wait staff and the maids)
n(B ∩ C) = 275 (number of guests who tip both the luggage handlers and the maids)
n(A ∩ B ∩ C) = 245 (number of guests who tip all three services)
n(U) = 2560 (total number of guests surveyed)
n(U - A ∪ B ∪ C) = 210 (number of guests who do not tip any of the services)
Using these values, we can calculate the number of guests who tip exactly two of the three services:
n(A ∩ B - A ∩ B ∩ C) = n(A ∩ B) - n(A ∩ B ∩ C)
= 755 - 245
= 510
n(A ∩ C - A ∩ B ∩ C) = n(A ∩ C) - n(A ∩ B ∩ C)
= 700 - 245
= 455
n(B ∩ C - A ∩ B ∩ C) = n(B ∩ C) - n(A ∩ B ∩ C)
= 275 - 245
= 30
Now, to find the total number of guests who tip exactly two of the three services, we sum up these values:
n(A ∩ B - A ∩ B ∩ C) + n(A ∩ C - A ∩ B ∩ C) + n(B ∩ C - A ∩ B ∩ C)
= 510 + 455 + 30
= 995
Therefore, 995 guests surveyed tip exactly two of the three services.