srivathsan wrote:In how many ways can 3 canadians, 2 americans and an indian can be arranged in a row such that all 3 canadians and 2 americans are always together?
Hmm, there's some ambiguity here.
Are you saying that the 3 Canadians must always sit together, and the 2 Americans must always sit together? (e.g., C,C,C,I,A,A would be fine)
Or, are you saying the 5 people who are either Canadian or American must sit together. (e.g., C,A,A,C,C,I would be fine)
I'm assuming that it's the first one. That is,
the 3 Canadians must always sit together, and the 2 Americans must always sit together? (e.g., C,C,C,I,A,A would be fine)
Take the task of seating everyone and break it into stages.
Stage 1: Arrange the 3 Canadians in a row.
We have a nice rule for this. We can arrange n unique objects in a row in n! ways.
So, we can arrange the 3 Canadians in 3! ways (
6 ways).
Stage 2: Arrange the 2 Americans in a row.
We can complete this stage in 2! ways (
2 ways).
Stage 3: Arrange the 1 Indian in a row.
We can complete this stage in (
1 way).
Stage 4: Take the pre-arranged
groups of Canadians, Americans and Indian, and arrange them in a row.
There are 3
groups, so we can arrange them in 3! ways (
6 ways).
By the Fundamental Counting Principle (FCP) we can complete all 4 stages (and thus seat all 6 people) in
(6)(2)(1)(6) ways ([spoiler]= 72 ways[/spoiler])
Cheers,
Brent
Aside: For more information about the FCP, we have a free video on the subject:
https://www.gmatprepnow.com/module/gmat-counting?id=775
