.
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 530 graduates
208 had at least one year of Spanish.
174 had at least one year of French.
144 had at least one year of German.
40 had at least one year of Spanish and French.
29 had at least one year of Spanish and German.
23 had at least one year of French and German.
5 had at least one year of all three languages.
(a) How many of the graduates had at least 1 yr of at least one of the three languages?
(b) How many of the graduates had at least 1 yr of exactly one of the three languages?
c) How many of the graduates had less than 1 yr of any of the three languages?
~~~~~~~~~~~~~~~~~~~~~~~
You are given
S = 208
F = 174
G = 144
SF = 40
SG = 29
FG = 23
SFG = 5
(a) How many of the graduates had at least 1 yr of at least one of the three languages?
There is a special formula to calculate n(S U F U G), when you know the numbers of elements in the subsets S, F, G
and in each of intersections SF, SG, FG, SFG:
n(S U F U G) = S + F + G - SF - SG - FG + SFG =
= 208 + 174 + 144 - 40 - 29 - 23 + 5 = 439.
Answer to question (a) is 439.
(b) How many of the graduates had at least 1 yr of exactly one of the three languages?
(S - SF - SG + SFG) + (F - SF - FG + SFG) + (G - SG - FG + SFG).
Substitute the given data and calculate.
c) How many of the graduates had less than 1 yr of any of the three languages?
Answer to this question is the difference 530 - 439,
where 530 came from the condition and 439 is the answer to question (a).
--------------------
If you need more explanations, look into the lessons
- Counting elements in sub-sets of a given finite set
- Advanced problems on counting elements in sub-sets of a given finite set
in this site.
