Homework SourseThe Department of Foreign Languages of a liberal arts colleg
The Department of Foreign Languages of a liberal arts college conducted a survey of its recent graduates to determine the foreign language courses they had taken while undergraduates at the college. Of the 510 graduates,
203 had at least one year of Spanish.
175 had at least one year of French.
147 had at least one year of German.
43 had at least one year of Spanish & French
26 had at least one year of Spanish & German
21 had at least one year of French & German
6 had at least one year of all three languages.
(a)How many of the graduates had at least 1 year of at least one of the three languages?
(b)How many of the graduates had at least 1 year of exactly one of the three languages?
(c) How many of the graduates had less than 1 year of any of the three languages?
Solution
we have given data and it can be written as
n(s) = 203
n(F) = 175
n(G) = 147
n(S F) = 43
n(S G) = 26
n(F G) = 21
n(S F G) = 6
a) How many of the graduates had at least 1 year of at least one of the three languages ?
n(At least one set) = n(S) + n(F) + n(G) – n(S F) – n(F G) – n(G S) + n(S F G)
n(At least one set) = 203 + 175 + 147 - 43 - 21 - 26 +6 = 441
So 441 graduates had at least 1 year of at least one of the three languages
(b)How many of the graduates had at least 1 year of exactly one of the three languages?
n(Exactly one set) = n(S) + n(F) + n(G) – 2*n(S F) – 2*n(F G) – 2*n(G S) + 3*n(S F G)
n(Exactly one set) = 203 + 175 + 147 - 86 - 42 - 52 + 18 = 363
So 363 graduates had at least 1 year of exactly one of the three languages
(c) How many of the graduates had less than 1 year of any of the three languages?
from queation (a) we have total number graduates have at least 1 year of at least one of the three languages.
so here it is asking less than 1 year and it should be considered as no set.
total number of students = n(no set) + n(At least one set)
510 = n(no set) + 441
n(no set) = 510-441 = 69
so 69 graduates had less than 1 year of any of the three languages
