18 votes 18 votes In a room containing $28$ people, there are $18$ people who speak English, $15$ people who speak Hindi and $22$ people who speak Kannada. $9$ persons speak both English and Hindi, $11$ persons speak both Hindi and Kannada whereas $13$ persons speak both Kannada and English. How many speak all three languages? $9$ $8$ $7$ $6$ Set Theory & Algebra gate1998 set-theory&algebra easy set-theory + – Kathleen asked Sep 25, 2014 edited Mar 26, 2021 by soujanyareddy13 Kathleen 6.7k views answer comment Share Follow See all 3 Comments See all 3 3 Comments reply Venkat Sai commented Dec 20, 2017 reply Follow Share the should mention that all the people know speaking some language else we can not assume the sample space 2 votes 2 votes mrinmoyh commented Aug 15, 2019 reply Follow Share yes 0 votes 0 votes HitechGa commented Jul 18, 2021 reply Follow Share This question is different from the question here. Here the venn diagram structure is flexible because no explicit values of $|V∩F|$,$|V∩E|$ or $|F∪E|$ are given, so we are free to think of any possible sort of overlapping and variations in the cardinality of the union of the three sets. But in this question, the cardinality of the intersection of pairs of sets are given so the structure of the venn diagram of the set as such is fixed and hence cardinality of union of the sets is also fixed. Now if the cardinality of the union of the sets is not equal to the number of people in the room then we cannot possibly give a specific answer to the question. So we assume that cardinality of the union of the sets is equal to the number of people i.e. each people speaks at least one of the three languages. 2 votes 2 votes Please log in or register to add a comment.
Best answer 23 votes 23 votes Apply set formula of $A$ union $B$ union $C$ $28 = (18 + 15 + 22) - (9 + 11 + 13) + x$ $28 = 55 - 33 + x$ $x = 6$ Correct Answer: $D$ Digvijay Pandey answered May 7, 2015 edited Apr 24, 2019 by Naveen Kumar 3 Digvijay Pandey comment Share Follow See all 4 Comments See all 4 4 Comments reply mehul vaidya commented Apr 12, 2018 reply Follow Share your answer is correct but solution is not 100% correct please refer Arjun's answer to https://gateoverflow.in/2314/gate1993_17 point is that it is not mentioned anywhere that all 28 people must speak atleast one of three language. hence answer could be either 0 ,1 ,2 ,3 ,4 ,5 ,6 , but as in option only 6 is given , hence answer is 6 13 votes 13 votes akash.dinkar12 commented Jul 26, 2018 i edited by akash.dinkar12 Jun 25, 2019 reply Follow Share mehul vaidya u are absolutely correct. Since the question is not giving any information about the people who don't speak any of the languages, we should not neglect it. $| E | = 18, | H | =15 , |K| =22 , |E∩ H| =9 , |E∩K| = 11 , |H∩K|, |E∩H∩K| =X$ Let the people who don't speak any of the languages is Y; According to the principle of Mutual Exclusion: $|E∪ H∪ K| = 18 + 15+ 22 -(9+11+13) + X$ $28 - Y = 22 + X$ $X + Y = 6 ----------(1)$ There are $7$ possible pairs of values which satisfy this equation here; $(6,0),(5,1)(4,2)(3,3)$ Since we have to follow the options, we will take pair$(6,0)$ indicates $X =6$ otherwise $X$ can be $5, 4,3,2,1,0$ 31 votes 31 votes Kuldeep Pal commented Dec 5, 2019 reply Follow Share @mehul vaidya did not get you bro 0 votes 0 votes Kuldeep Pal commented Dec 5, 2019 reply Follow Share @mehul vaidyadid not get you bro 0 votes 0 votes Please log in or register to add a comment.
2 votes 2 votes In question given that, n(EUHUK) =28 ,n(E∩H) =9 ,n(E∩K) =11 ,n(H∩K) =13 ,n(E)=18 ,n(H)=15 , n(K)=22 n(EUHUK) = n(E) + n(H)+ n(K) - [ n(E∩H) + n(E∩K) + n(H∩K) ] + n(E∩H∩k) 28 = 18 +15 +22 -[9+11+13] + n(E∩H∩k) n(E∩H∩k) = 6 Option (D)6 ,is correct. Warrior answered Jul 29, 2017 Warrior comment Share Follow See all 0 reply Please log in or register to add a comment.
2 votes 2 votes Using Venn diagram Nithin2698 answered Jun 18, 2019 Nithin2698 comment Share Follow See 1 comment See all 1 1 comment reply Anurag Tiwari 1 commented Oct 15, 2019 reply Follow Share nice explanation 0 votes 0 votes Please log in or register to add a comment.
0 votes 0 votes Let A be the people who speaks English. Let B be the people who speaks Hindi. Let C be the people who speaks Kannada. (A∪B∪C) = A + B + C - (A∩B) - (B∩C) - (C∩A) + (A∩B∩C) 28 = 18 + 15 + 22 - 9 - 11 - 13 + (A∩B∩C) ∴ (A∩B∩C) = 6 varunrajarathnam answered Oct 3, 2020 varunrajarathnam comment Share Follow See 1 comment See all 1 1 comment reply CheeseCuBES commented Jan 29, 2021 i edited by CheeseCuBES Jan 29, 2021 reply Follow Share by venn diagram Total – None = a + b = c – (ab +bc + ca) + (abc) 28 – none = 18 + 15 + 22 -(9+11+13) + (abc) (abc)- none = 6 0 votes 0 votes Please log in or register to add a comment.