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Combinations With Constraints


audi19

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Is there a better way to understand combinations with constraints? For example, I'm reading the Manhattan GMAT prep Word Translations and I am trying to understand the explanation to this question but can't quite seem to get it.

 

G, M, P, J, B, and C can sit next to each other in 6 adjacent seats. If M and J will not sit next to each other, how many different arrangements can the six people sit?

 

Manhattan's explanation:

 

1) Find the number of ways in which six people can sit in 6 chairs: 6!= 720

2) Find the number of combinations in which M & J are sitting next to each other.

a. There are 2! =2 ways in which M&J can be arranged and 4!=24 ways in which the other 4 can be arranged.

b. Therefore there are 2x24=48 permutations for each seat pair.

c. Since there are 5 seat pairs, there are 5x48=240 permutations in which M&J are sitting next to each other.

3) The number of permutations that they are not seated next to each other is 720-240 = 480.

 

Is there a shortcut to this?

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  • 5 years later...
I was afraid of that response. Thanks for your response!

 

The easier approach would be to consider M and J as one unit.

Now, consider the problem where there are 5 places and 5 ppl to be places in those five places. The number of permutations is 5! = 120. M and j although always beside each other can be arranged among themselves in 2!= 2 ways. So total permutations with M and J beside each other = 120 * 2 = 240.

 

The answer is:

Reduce 240 from total number of permutations of arranging 6 people on 6 places i.e. 6!=720

i.e. answer is 720-240 =480

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  • 1 month later...
The easier approach would be to consider M and J as one unit.

Now, consider the problem where there are 5 places and 5 ppl to be places in those five places. The number of permutations is 5! = 120. M and j although always beside each other can be arranged among themselves in 2!= 2 ways. So total permutations with M and J beside each other = 120 * 2 = 240.

 

The answer is:

Reduce 240 from total number of permutations of arranging 6 people on 6 places i.e. 6!=720

i.e. answer is 720-240 =480

 

This does seem like a faster approach. Thank you!

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