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In how many different ways can 6 identical belts

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In how many different ways can 6 identical belts [#permalink] New post 08 Oct 2019, 05:27
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Question Stats:

40% (02:48) correct 60% (01:40) wrong based on 10 sessions
In how many different ways can 6 identical belts and 5 identical hats be distributed among 8 different children,
so that each child receives at least 1 item, no child receives 2 or more belts, and no child receives 2 or more hats?

A) 240
B) 256
C) 420
D) 480
E) 560
[Reveal] Spoiler: OA

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Re: In how many different ways can 6 identical belts [#permalink] New post 09 Oct 2019, 08:30
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GreenlightTestPrep wrote:
In how many different ways can 6 identical belts and 5 identical hats be distributed among 8 different children,
so that each child receives at least 1 item, no child receives 2 or more belts, and no child receives 2 or more hats?

A) 240
B) 256
C) 420
D) 480
E) 560


The first thing we need to do is determine how many children fall into each category (e.g., receive a hat but no belt, receive a belt but no hat, or receive both a hat and a belt)

To do so, we can use the Double Matrix Method. This technique can be used for most questions featuring a population in which each member has two characteristics associated with it (aka overlapping sets questions).
Here, we have a population of children, and the two characteristics are:
- receives a hat or doesn't receive a hat
- receives a belt or doesn't receive a belt

When we apply the Double Matrix Method, the distribution of the 8 children looks like this:
Image
ASIDE: When it comes to populating the matrix, the key piece of information is that the question tells us that each child receives at least 1 item, which means there are ZERO children in the bottom right box (indicating those children who received neither a hat nor a belt)

If you want to learn more about the Double Matrix Method, watch the video below.

Okay, once we've determined the number of children who fall into each category, it's simply a matter of choosing the children for each category.
We'll do so in stages

Stage 1: Select 3 children to receive both a hat and a belt
Since the order in which we select the children does not matter, we can use combinations.
We can select 3 children from 8 children in 8C3 ways (56 ways)
So, we can complete stage 1 in 56 ways

Stage 2: Select 2 children to receive a hat but no belt
There are now 5 children remaining.
Once again, we'll use combinations (since the order in which we select the children does not matter)
We can select 2 children from the remaining 5 children in 5C2 ways (10 ways)
So, we can complete stage 2 in 10 ways.

Stage 3: Select 3 children to receive a belt but no hat
There are now 3 children remaining.
We can select 3 children from the remaining 3 children in 3C3 ways (1 way)
So, we can complete stage 3 in 1 way.

By the Fundamental Counting Principle (FCP), we can complete all 3 stages (and thus distribute all of the hats and belts) in (56)(10)(1) ways (= 560 ways)

Answer: E

Cheers,
Brent

RELATED VIDEO FROM MY COURSE
[you-tube]https://youtu.be/ZVkLdDVV5Bs[/you-tube]
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Re: In how many different ways can 6 identical belts [#permalink] New post 02 Dec 2019, 13:57
Time-wise at the actual GRE test, is this type of exercise worth solving? Is there any way to approximate a solution and save time? Sure, a math expert might come up with the mathematical model, write and solve all the combinatorics in under 3 minutes. But for all of the other simple mortals out there (like me), is there a simpler way?
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Re: In how many different ways can 6 identical belts [#permalink] New post 05 Dec 2019, 07:49
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danieldira wrote:
Time-wise at the actual GRE test, is this type of exercise worth solving? Is there any way to approximate a solution and save time? Sure, a math expert might come up with the mathematical model, write and solve all the combinatorics in under 3 minutes. But for all of the other simple mortals out there (like me), is there a simpler way?


While this is a super difficult question (165+), the concepts tested are within the scope of the GRE.
Once one recognizes that this is an overlapping sets question, it shouldn't take longer than 15 or 20 seconds to complete the Double Matrix, at which point we need only examine three stages in order to apply the fundamental counting principle.

Approximating the answer won't really help us here because several of the answer choices are relatively close together.
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Re: In how many different ways can 6 identical belts [#permalink] New post 07 Dec 2019, 10:37
E
Re: In how many different ways can 6 identical belts   [#permalink] 07 Dec 2019, 10:37
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In how many different ways can 6 identical belts

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