What is the maximum possible number of these people who can speak only one language?

Question

At an international conference of $ \ 100 \ $ people, $ \ 75 \ $ speak English, $ \ 60 \ $ speak Spanish and $ \ 45 \ $ speak Swahili (and everyone present speaks at least one of these languages).

(i) What is the maximum possible number of these people who can speak only one language? In this case how many people speak only English, how many speak only Spanish, how many speak only Swahili, and how many speak all three?

Answer:

Let

Total number of people $ \ =100 \ $ ,

$ A= \ $ number of people speaking English,

$ B= \ $ number of people speaking Spanish,

$ C= \ $ number of people speaking Swahili,

By Given condition,

$ A=75, \ B=60, \ C=45 \ $

$ n(A \cup B \cup C)=100 \ $

But how to get next step to answer the given questionS?

Help me doing this.

Answer 1

Since every one speaks at least one of these languages, Inclusion-Exclusion says that $$ 100=\overbrace{|E|}^{75}+\overbrace{|S|}^{60}+\overbrace{|W|}^{45}-|E\cap S|-|E\cap W|-|S\cap W|+|E\cap S\cap W|\tag1 $$ This means that $$ |E\cap S|+|E\cap W|+|S\cap W|-|E\cap S\cap W|=80\tag2 $$ Number who speak only English: $$ |E|-|E\cap S|-|E\cap W|+|E\cap S\cap W|\tag3 $$ Number who speak only Spanish: $$ |S|-|E\cap S|-|S\cap W|+|E\cap S\cap W|\tag4 $$ Number who speak only Swahili: $$ |W|-|E\cap W|-|S\cap W|+|E\cap S\cap W|\tag5 $$ $(3)$, $(4)$, and $(5)$ say that the number who speak only one language is $$ |E|+|S|+|W|-2|E\cap S|-2|E\cap W|-2|S\cap W|+3|E\cap S\cap W|\tag6 $$ Using $(1)$ and $(2)$, $(6)$ becomes $$ 100-80+|E\cap S\cap W|\tag7 $$ So we need to maximize $|E\cap S\cap W|$.

Since $|E\cap S|,|E\cap W|,|S\cap W|\ge|E\cap S\cap W|$, $(2)$ sets the limit $$ |E\cap S\cap W|\le40\tag8 $$ $(8)$ can be achieved:

By $(7)$ and $(8)$, the maximum number of people who only speak one language is $60$.

Answer 2

My first thought was more or less identical to turkeyhundt's comment:

It is my intuition that the way to maximize the number of single-language-speakers would be to also maximize the number of three-language-speakers. In that case, 40 three language speakers would be needed to leave 35 English-only, 20 Spanish-only, and 5 Swahili-only. Just a guess and not a proof though

To convert this intuition into a proof, we need some way to relate the number of single-language-speakers to the number of three-language-speakers.

At this point, if I weren't typing on a computer, I'd probably draw a Venn diagram. Here's one way to think about it.

We know $$ 100 = N_1 + N_2 + N_3$$

Where $N_i$ is the number of people speaking $i$ languages.

Based on our intuition, we want to remove $N_2$ from that equation.

Let $a,b,c$ be the number of people speaking each language only, so $N_1 = a+b+c$. Then we know $75-a$ people speak language A and some other language(s), $60-b$ speak B and some other, and $45-c$ speak C and some other. That means the total is $180-(a+b+c)=180-N_1$, but that double-counts the two-language speakers and triple-counts the three-language speakers, so $N_2$ = $\frac{180-N_1-3N_3}{2} = 90 - \frac12N_1 - \frac32N_3$. Plugging that in gives $$100 = N_1+90 -\frac12N_1 - \frac32N_3 + N_3 = 90 + \frac12N_1 -\frac12N_3$$ and rearranging gives $$N_1 = N_3 + 20$$

Now we can proceed along the lines turkeyhundt suggested. Maximizing $N_3$ clearly maximizes $N_1$, but if $N_3 >40$, then $N_1>60$ and so $N_1+N_3>100$, which is impossible. We check $N_3=40$ works, giving maximum $N_1=60$.

Answer 3

My answer to your question is $ n(A)$+$n(B)$ + $n(C)$ -$n(A \cup B)$-$n(B \cup C)$-$n(A \cup C)$+ $n(A \cup B \cup C)$ Take respective events as $A, B, C$ and using the above form you should be getting your answer.