The document Knots & Gaps GMAT Notes | EduRev is a part of the GMAT Course Quantitative Aptitude for GMAT.

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Often we come across probelms in Permutations and combinations in which we are given a condition that two or more entities must be together, or that they cannot be together. We will try to look into those cases here.

Situation 1: 10 people are seated in a row, but A and B must be together, so in how many ways can we arrange them in a row.

In this case we tie a knot to A and B together, and treat them as 1 person first.

Next, we see the total number of people left, in this case those are 8.

So the total number of entities are 9.

Now these 9 entities can be seated in 9! ways.

But A and B, can be seated as AB or BA, hence we multiply the above number with 2!.

And our answer to the above problem is 9!*2!

Now we move on to "Gaps"

Situation 2: 10 people are seated in a row, but A and B cannot be together.

Now we know that A and B cannot be together, so let's talk about the other people first.

Those people can take 8 seats in 8! ways.

When those people take 8 seats- gaps are created as follows:

Gap - G

People- N

**G **N3 **G **N4 **G **N5 **G **N6 **G** N7 **G** N8 **G** N9 **G** N10 **G**

Now when you count the number of gaps among the 8 remaining people, excluding A and B, there are 9 gaps

A and B can take place in 9 of those gaps.

We can select those gaps in 9C2 ways.

And those two people can also be seated in 2! ways.

So the answer to situation 2 is

8! * 9C2 * 2!

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