[Aptitude] PnC: arrange such that two people Always sitting together?

Q. How many ways can four children be made to stand in a line such that two of themA and B are always together? (-asked in UPSC Prelims 2010)

Confused? You wont be, after reading this.DONOT Proceed further, without reading The First article on basic concept ofPermutation and Combination

Always Sit together1.Never Sit together2.Demo Questions: Permutation sitting together3.Always Select4.Never Select5.Demo Question: Selection6.

CASE: ALWAYS SIT TOGETHER

Three couples are standing in the compound of Gokuldham society

Jethalala + DayabenMehta saab + AnjaliAiyyer + Babita-ji

How many ways can you arrange them in such a way that Jethalal and Dayaben are

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always sitting together?First, ask yourself, is it a permutation problem or a combination problem?Since sitting= order matters, this is a permutation problem.Now let us consider Jethalal and Daya as ONE Person. And theyll sit in one chair.

Now youve the case of 5 Persons and 5 Chairs.How many ways can use arrange them?As we saw in the previous article= 5 x 4 x 3 x 2 x 1=5! Factorial Ways.=120But here order matters, this is a permutation problem soIn the sitting order Daya-Jetha (DJ) and Jetha-Daya (JD) are two differentarrangements.So now time to split that One person Apart.How many ways can use sit 2 persons (DJ) in 2 chairs?2 x 1=2 ways.So, here weve5 people sit in 5! Ways (=120)2 people (Daya-Jetha) can sit in 2 ways.According to fundamental counting120 x 2 = 240 ways you can arrange this.

(Formula: 2P2 x 5P5); but we dont need to use it.

CASE: DONT SIT TOGETHER

Que. Same 3 couples (=6 persons), 6 chairs. How many ways you can arrange them sothat Dayaben and Jethalal (DJ) never sit together.Ans.First how many ways can 6 people sit in 6 chairs?

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=6! Factorial Ways. (=720) ; Solved in the first article.

How many ways can 6 people sit such that Dayaben and Jethalal (DJ) always sittogether=240 ways ; solved in previous case.

SoTotal ways = Cases where DJ together + Cases where DJ not together720=240 + DJ not togetherDJ not together = 720-240=480 ways.Answer: in 480 ways you can arrange the given 6 people such that Dayaben andJethalal dont sit together.Formula: 6P6 minus [2P2 x 5P5]; but we dont need to use it.

Demo Question:Sitting together

In CSAT Paper II (Aptitude), this can appear under different situations

How many ways can four children be made to stand in a line such that two ofthem A and B are always together? (-already asked in UPSC Prelims 2010)How many ways can you arrange the word Trial where letters T and L arealways together?How many ways can you arrange the word Trial where letters R and A arenever together?

COMBINATION PROBLEM: ALWAYS SELECT, NEVER SELECT

CASE: ALWAYS SELECT

Director Asit Modi wants to shoot an episode in Kutch. Out of the 6 characters, howmany ways can he select 4 character such that Dayaben and Jethalal (DJ) are always inthe crew?

Here order doesnt matter, only selection matters. After all this is just anotherCommitteeDJ and JD are one and same.Now deconstruct the problem6 peopleSelect 4.2 must be selected (DJ)

How many ways can you select Dayaben and Jethalal? : One way.Matter finish.Now worry about the remaining 2 vacancies.weve 4 characters left (Mehta-Anjali, Aiyyar-Babita)And weve to select two of them.Classical combination problem. Solve it according to previous article.

(4*3)/(2*1)=6 ways.

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Fundamental Counting=(Daya Jetha selection ways) x (2 out of 4 selection)=1 x 6=6 ways.Final Answer : Asit Modi can select the characters in 6 ways such that Jethalal,Dayaben are always in the crew. In case youre curious, here is the graphicalrepresentation of it

CASE: NEVER SELECT in the crew

Q. Select 4 out of 6 given characters for Kutch-episode, such that Dayaben and Jethalalare never selected in the crew?

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Ans.Very easy. Drop DJ, how many left?=4 Characters: Mehta-Anjali, Aiyyar-Babita4 Characters, 4 vacancies. How many ways can you select? Just one way: youll haveto select all of them! because order doesnt matter, this is a combination problem!

PS. If you wish to solve it by formula: it is 4C4=1; because nCr=1 when n=r.

Demo Questions for CSAT paper II (Aptitude)

A zoo is has to purchase 3 animals out 5 given animals: Zebra, Elephant, Tiger,Lion and Leopard. How many ways can they do it? (5c3)^Such that Zebra and Elephant are always selected? (3C1)^Such that Tiger is never selected? (4C3)

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Posted By Mrunal On 07/03/2012 @ 19:37 In the category Aptitude

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