(a) All three languages - 9
(b) Indonesian only - 10
(c) none of the languages - 12
(d) At least one language - 168
(e) Either one or two of the three languages - 159
Step-by-step explanation:
The question require the knowledge of Set theory and its Formulas.
Total pupils = 180
French (Let F) = 110
German (Let G) = 88
Indonesian (Let I) = 65
French and German (F intersection G) = 40
German and Indonesian (G intersection I) = 38
French and Indonesian (F intersection I) = 26
German only = 19
(a) All three languages
We are given only German speaking people are 19
Only German = n(G) - n( F intersection G) - n(G intersection I) + n(F intersection G intersection I)
19 = 88 - 40-38 + n(G intersection I) + n(F intersection G intersection I)
n(G intersection I) + n(F intersection G intersection I) = 9
n(G intersection I) + n(F intersection G intersection I) represents the number of pupils speaking who study all the three languages.
(b)Indonesian only
n( I) - n(G intersection I) + n(F intersection I) + n(G intersection I) + n(F intersection G intersection I)
65 - 38 - 26 + 9 = 10
So 10 pupils speak Indonesian only
(c)none of the languages
It will be equal to Total - pupils speaking any of the three languages
Any of the three languages = (F union G union I)
= n(F) +n(G) + n(I) -n(F intersection G) - n(G intersection I) -n(F intersection I)
+ n(F intersection G intersection I)
n(G intersection I)
= 110 + 88 + 65 -40 -38-26 + 9
= 263 - 95
= 168
So 168 pupils speak any of the three languages
None speakers = 180 - 168 = 12 pupils
So 12 pupils do not speak any of the three languages.
(d)at least one language
At least one language has the meaning that the person can either speak one two or all three languages, so it will be same as we proceeded above
Any of the three languages = (F union G union I)
= n(F) +n(G) + n(I) -n(F intersection G) - n(G intersection I) -n(F intersection I)
+ n(F intersection G intersection I)
n(G intersection I)
= 110 + 88 + 65 -40 -38-26 + 9
= 263 - 95
= 168
(e) either one or two of the three languages
Pupils can speak one or two language but not all the three so will subtract all the three language speaker from the total.
Total speaker = 168
All three languages speaker = 9
Either one or two of the three languages speaker = 168 - 9 = 159
