Six friends go to see a comedy at the local movie theater and sit in a

Question

Six friends go to see a comedy at the local movie theater and sit in a particular row together. If Grady and Howard are among those 6 friends but can never sit next to each other, how many 6-person seating arrangements are possible?

A. 360
B. 420
C. 480
D. 600
E. 720

A. 360 
B. 420 
C. 480 
D. 600 
E. 720

davedekoos · Answer

Ways to arrange 6 people in a row = 6! = 720

From this we want to subtract all the arrangements where Grady and Howard are next to each other. If we consider Grady and Howard one unit (GH), then the number of arrangements of the GH unit and the 4 other friends is 5! And the number of arrangements of G and H within the GH unit are 2 (GH or HG). So the total number of arrangements of the 6 friends with Grady and Howard together are 5!*2

Number of permutations where Grady and Howard are not seated next to each other = 6! - 5!*2 = 720-120*2 = 480

Answer: C

rhine29388 · Answer

6 persons can be arranged in a row in 6! ways = 720 ways
consider Grady and Howard as 1 group in which they both sit together so we have 4 persons and 1 group
they can be arranged in 5! ways
Grady and Howard can arrange themselves in 2! ways
so in total the number of ways in which they are together is 5! * 2! = 120 * 2 = 240
subtract this number from total number of arrangements to get the number of arrangements in which Grady and Howard are not sitting together
720 - 240 = 480
Correct Answer - C

Senthil1981 · Answer

If Gary sits on one of the two corners, then Hardy will have 4 options, So 2(Gary's 2 corner) * 4(Hardy's 4 position) *  4! , the remaining 4 can have 4! arrangement,
And if Gary sits other than the corner, then Hardy will have only 3 options, So that is 4 * 3 * 4!

In total 2 * 4 * 4! + 4 *3 * 4! = 480.

answer is C.

Archit3110 · Answer

6 friends can sit in 6! ways and if 2 decide not to sit together ; 5! *2! 
total ways ; 6!- ( 5!*2!) = 480
IMO C

Feb2024 · Answer

Hello Experts  KarishmaB   Bunuel   chetan2u   gmatophobia   MartyMurray   IanStewart

Although I got the answer correct on the first attempt, I wonder why can we not use the logic that in exactly half of the cases, the two kids will be seating next to each other, and so required number of seating arrangements = 6!/2 = 360. What is wrong here in my understanding?

IanStewart · Answer

I'm curious why you concluded they would sit together half the time? If that were true, your logic would be perfect, but they do not sit together half the time -- they sit together only 1/3 of the time (there are 5 pairs of seats that are next to each other, and 6C2 = 15 pairs of seats in total, so 5/15 = 1/3 of the time if you choose two random seats, those seats are next to each other).

chetan2u · Answer

There are six of them, and let us say in half cases E sits with A.
But similarly E will be sitting with B, C, D etc in half cases each. But that will add up to more than total even if you take E having two persons on either side.

You could use this logic if we were finding whether A was sitting on left side of E or some other similar situation.

Bunuel · Answer

You're likely confusing this setup with a different type of question, one where you're asked in how many arrangements Grady is to the left of Howard (not necessarily next to him). In that case, yes, exactly half of the 6! arrangements would have Grady to the left and the other half to the right, so dividing by 2 works.

However, here we're dealing with how many arrangements do not have Grady and Howard sitting next to each other, which is different. For example, with 2 people, they'd sit next to each other in 100% of the cases. With 3 people, there are 3! = 6 arrangements, and in 4 of them Grady and Howard would be adjacent, so 4/6 = 2/3 = 66% of the time. With 4 people, there are 24 arrangements, and in 12 of them they'd be next to each other, so exactly 50%. As the number of people increases, the proportion of cases where they sit next to each other decreases.

Hope it's clear.

Feb2024 · Answer

IanStewart   chetan2u   Bunuel

Thank You to all the experts for responding such quickly and each one providing their own lessons. 
The concept is clearer now 😚

Six friends go to see a comedy at the local movie theater and sit in a

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