The question seems to be asking:
How many ways can Bob and Rachel be seated in a row of 5 chairs if Bob must sit to the left of Rachel and the other 3 chairs are to remain empty?
TOTAL WAYS to arrange Bob and Rachel in the 5 chairs:
Number of options for Bob = 5. (Any of the 5 chairs.)
Number of options for Rachel = 4. (Any of the 4 remaining chairs.)
To combine these options, we multiply:
5*4 = 20.
Total ways to seat Bob TO THE LEFT of Rachel:
In half of the 20 arrangements counted above, Bob will be to the left of Rachel; in the other half, Rachel will be to the left of Bob.
Thus, we divide by 2:
20/2 = 10.
Another option is to write out the acceptable arrangements.
Let B = Bob, R = Rachel, X = an empty chair.
Here are all of the acceptable arrangements:
BRXXX
BXRXX
BXXRX
BXXXR
XBRXX
XBXRX
XBXXR
XXBRX
XXBXR
XXXBR
Total ways = 10.
Beat the probability/combinatorics Qs
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Here's one more approach.sachindia wrote:There are 5 chairs. Bob and Rachel want to sit such that Bob is always left to Rachel. How many ways it can be
done?
Select any 2 chairs out of the 5 chairs.
Since the order of the selected chairs does not matter, we can use combinations.
We can select 2 chairs in 5C2 ways (10 ways)
At this point we're done. These 2 chairs are for Bob and Rachel, and if Bob must sit to the left of Rachel, there's only one way for them to occupy these 2 chairs.
Answer = 10
Aside: If anyone is interested, we have a free video on calculating combinations (like 5C2) in your head: https://www.gmatprepnow.com/module/gmat-counting?id=789
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sachindia
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Hi Brent,These 2 chairs are for Bob and Rachel, and if Bob must sit to the left of Rachel, there's only one way for them to occupy these 2 chairs.
I didn't understand the above.. Please elaborate.
Regards,
Sach
Sach
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sachindia
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Also the nCr formula can be applied when the ordering doesn't matter. Here, the ordering does matter,right? Bob must sit to the left of Rachel.
Regards,
Sach
Sach
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Here's another way to look at it.
Take the task of seating Bob and Rachel and break it into stages.
Stage 1: Select two chairs
The order of the two selected chairs doesn't matter. For example, selecting chair #3 then chair #5 is the same as selecting chair #5 then chair #3
So, we can use combinations.
We can select 2 chairs in 5C2 ways (10 ways)
Stage 2: Seat Bob and Rachel in the two chairs
Since Bob must sit to the left of Rachel, there's only 1 way for them to occupy these 2 chairs.
In other words, there's only 1 way to accomplish stage 2
By the Fundamental Counting Principle (FCP) we can complete the two stages (and thus seat Bob and Rachel) in (10)(1) ways ([spoiler]= 10 ways[/spoiler])
Does that help clarify things?
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
Take the task of seating Bob and Rachel and break it into stages.
Stage 1: Select two chairs
The order of the two selected chairs doesn't matter. For example, selecting chair #3 then chair #5 is the same as selecting chair #5 then chair #3
So, we can use combinations.
We can select 2 chairs in 5C2 ways (10 ways)
Stage 2: Seat Bob and Rachel in the two chairs
Since Bob must sit to the left of Rachel, there's only 1 way for them to occupy these 2 chairs.
In other words, there's only 1 way to accomplish stage 2
By the Fundamental Counting Principle (FCP) we can complete the two stages (and thus seat Bob and Rachel) in (10)(1) ways ([spoiler]= 10 ways[/spoiler])
Does that help clarify things?
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
Brent Hanneson - Creator of GMATPrepNow.com
