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In how many ways can we put 4 different balls in 3 different boxes

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by BTGModeratorVI » Mon Apr 13, 2020 3:42 pm

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B

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In how many ways can we put 4 different balls in 3 different boxes when any box can contain any number of balls?

A. 80
B. 81
C. 64
D. 63
E. 82

Answer: B
Source: Jamboree
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Re: In how many ways can we put 4 different balls in 3 different boxes

by swerve » Tue Apr 14, 2020 7:45 am
BTGModeratorVI wrote: ↑
Mon Apr 13, 2020 3:42 pm
 In how many ways can we put 4 different balls in 3 different boxes when any box can contain any number of balls?

A. 80
B. 81
C. 64
D. 63
E. 82

Answer: B
Source: Jamboree
Since there are \(3\) boxes, we have

\(3\) possibilities for the first ball
\(3\) possibilities for the second ball
\(3\) possibilities for the third ball
\(3\) possibilities for the fourth ball

So, the total number of possibilities is \(3 \cdot 3 \cdot 3 \cdot 3 = 81 \Longrightarrow \)B
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Re: In how many ways can we put 4 different balls in 3 different boxes

by Brent@GMATPrepNow » Tue Apr 14, 2020 2:17 pm
BTGModeratorVI wrote: ↑
Mon Apr 13, 2020 3:42 pm
 In how many ways can we put 4 different balls in 3 different boxes when any box can contain any number of balls?

A. 80
B. 81
C. 64
D. 63
E. 82

Answer: B
Source: Jamboree
Take the task of distributing the 4 different balls and break it into stages.

Stage 1: Select a box for the 1st ball to go into.
There are 3 available boxes, so we can complete stage 1 in 3 ways

Stage 2: Select a box for the 2nd ball to go into.
There are 3 available boxes, so we can complete stage 2 in 3 ways

Stage 3: Select a box for the 3rd ball to go into.
There are 3 available boxes, so we can complete stage 3 in 3 ways

Stage 4: Select a box for the 4th ball to go into.
There are 3 available boxes, so we can complete stage 4 in 3 ways

By the Fundamental Counting Principle (FCP), we can complete all 4 stages (and thus distribute all 4 balls) in (3)(3)(3)(3)(4) ways (= 81 ways)

Answer: B

Note: the FCP can be used to solve the MAJORITY of counting questions on the GMAT. For more information about the FCP, watch this free video: https://www.gmatprepnow.com/module/gmat- ... /video/775

You can also watch a demonstration of the FCP in action: https://www.gmatprepnow.com/module/gmat ... /video/776

Then you can try solving the following questions:

EASY
- https://www.beatthegmat.com/what-should- ... 67256.html
- https://www.beatthegmat.com/counting-pro ... 44302.html
- https://www.beatthegmat.com/picking-a-5- ... 73110.html
- https://www.beatthegmat.com/permutation- ... 57412.html
- https://www.beatthegmat.com/simple-one-t270061.html


MEDIUM
- https://www.beatthegmat.com/combinatoric ... 73194.html
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- https://www.beatthegmat.com/digits-numbers-t270127.html
- https://www.beatthegmat.com/doubt-on-sep ... 71047.html
- https://www.beatthegmat.com/combinatoric ... 67079.html


DIFFICULT
- https://www.beatthegmat.com/wonderful-p- ... 71001.html
- https://www.beatthegmat.com/permutation- ... 73915.html
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- https://www.beatthegmat.com/no-two-ladie ... 75661.html
- https://www.beatthegmat.com/combinations-t123249.html


Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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Re: In how many ways can we put 4 different balls in 3 different boxes

by Scott@TargetTestPrep » Fri Apr 17, 2020 10:34 am
BTGModeratorVI wrote: ↑
Mon Apr 13, 2020 3:42 pm
 In how many ways can we put 4 different balls in 3 different boxes when any box can contain any number of balls?

A. 80
B. 81
C. 64
D. 63
E. 82

Answer: B
Source: Jamboree
To solve this problem, consider the balls, not the boxes. Ball #1 has 3 boxes into which it can be placed; ball #2 has 3 boxes into which it can be placed; ball #3 also has 3 boxes, as does ball #4.

Thus, the number of ways 4 different balls can be placed in 3 different boxes is 3^4 = 81.

Answer: B

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