Who Can Speak All 4 Languages?

The total number of journalists = 200.

People who speak German = 175 -> people who do not speak German = 25.

People who speak French = 150 -> people who do not speak French = 50.

People who speak English = 180 -> people who do not speak English = 20.

People who speak Japanese = 160 -> people who do not speak Japanese = 40.

The set of journalists that will have the MINIMUM number of people who know all 4 languages, is also the one that will have the MAXIMUM number of people who do not know at least 1 language.

And the total number of people who do not know at least 1 language will be MAXIMUM, when the sets of people not knowing a particular language are mutually exclusive.

Hence, the Maximum number of people who do not know at least 1 language = 25 + 50 + 20 + 40 = 135.

Thus, the MINIMUM number of people who know all 4 languages = 200 - 135 = 65.

There are a total of 150 + 160 + 175 + 180 = 665 languages spoken and 200 people. To minimize the number of people that speak all 4 everyone else should speak 3. If everyone spoke 3 languages there would be 3 x 200 = 600 languages spoken. There are actually 65 more languages, so 65 people speak all 4 languages.

Let us start with the min number of people speaking a language. So 150 people speak french. So remaining 50 people don't speak French. 160 speak Japanese. So Using Pigeon Hole Principle, it is true that 160-50 = 110 people speak both Japanese and French. So 90 doesn't speak Japanese and French both

175 speak German. So 175-90 = 85 speak Japanese, French and German. Thus 115 people doesn't speak these three languages.

180 speak English. So 180-115 = 65 speak all four languages.

As written in question- 175 people can speak German, -25 person can't speak german (so 25 people eliminated) 150 people can speak French, -50 person can't speak French-(50 people eliminated) 180 people can speak English, 20 can't speak english (20 eliminated) 160 people can speak Japanese. 40 can't speak japanese(40 eliminated)

Total people who are able of speaking every language is-200-(25+50+20+40) = 200-135 = 65

Let $A_1,A_2,A_3,A_4$ represent the sets of the people who speak German, French, English, and Japanese, and let $\begin{aligned} x & = |A_1 \cap A_2| + |A_1 \cap A_3| + |A_1 \cap A_4| + |A_2 \cap A_3| + |A_2 \cap A_4| + |A_3 \cap A_4| \\ y & = |A_1 \cap A_2 \cap A_3| + |A_1 \cap A_2 \cap A_4| + |A_1 \cap A_3 \cap A_4| + |A_2 \cap A_3 \cap A_4| \\ z & = |A_1 \cap A_2 \cap A_3 \cap A_4| \end{aligned}$ Then by the principle of inclusion and exclusion, $200 = (175 + 150 + 180 + 160) - x + y - z \implies 465 = x - y +z .$ Notice that we can also calculate the number of people who know at least two languages. First we start with $x$ people. We see that a person who knows 3 languages is included $\binom 32 = 3$ times. So we must subtract out $2y$ to get $x - 2y$ . Now, a person who knows 4 languages is included $\binom42 - 2\cdot \binom 43 = -2$ times. So we have $x - 2y + 3z$ , which is less than 200. Similarly, we can calculate that $200 \ge y - 3z$ . So, we have $\begin{aligned} 465 & = x - y + z \\ 200 &\ge x - 2y + 3z \\ 200 &\ge y - 3z \\ \end{aligned}$ We multiply $(2)$ by -1 and add $(1)$ to it to get $265 \le y - 2z$ . Now we add this to -1 times $(3$ to get $65 \le z$ . To verify that this works, we calculate $x,y,z$ with $\boxed{z = 65}$ .

Shouldn't it be 'maximum number of journalists who can speak all the 4 languages'? Because, there are at least 135 persons who can not speak either of the languages.

min(G and F)=125

min(G and F and E)=105

min(G and F and E and J)=65

the journalists =200 180 speak English so 20 do not speak it. because we want the min number of journalists that speak the 4 languages we can say that those 20 ones speak German and the others that speak German (175 - 20) speak English and German so 155 speak English and German . so ;now we have 20 people speak German only and 25 speak English only(180 - 155). and so on we have 45 speak Japanese who speak either English or German and by subtracting there are 115 people speak English , German and Japanese. we have 150 journalists speak French :20 of them speak German only , 25 speak English only and 40 speak English and German .now let all of them speak french so 150 - 85 = 65 who seak the 4 languages

no. of german spekers=g=175 no. of french speaking people =f=150 no. of english speaking people=e=180 no. of japanese speakin people=j=160

min no. of people speaking both german and french=min(g+f)=125 i.e. 175-(200-150) min(f+e)=130 min(e+j)=140

min(f+e+j)=90( i.e. 130-(180-140)) min(f+e+g)=105(i.e. 130-(150-125)

min(f+e+g+j)=65(i.e 105-(130-90))

Total number of people = 200

Out of these 150 can speak French and 160 can speak Japanese. Taking the extreme case, 50 speak only Japanese, 40 speak only French and 110 speak both French and Japanese. 110 people(Group A) speak both these languages and 90(Group B) only one of these two.

We have 175 people who can speak German and 180 people who can speak English. To minimise the number of people speaking all four languages, let us put 90 from both these groups in Group B. Group B now consists of people speaking three languages : English and German and French/Japanese.

Now we have 85 people who can speak German and 90 who can speak English. These have to be placed in Group A(110 people) appropriately to arrive to the solution.

Again taking the extreme case, we have 25 people speaking only English, 20 people speaking only German and 65 people speaking both German and English in this group of 110(Group A).

Therefore, people speaking all four languages = 65.