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

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Intern
Joined: 10 Sep 2016
Posts: 15
In how many ways can we put 4 different balls in 3 different boxes whe  [#permalink]

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Updated on: 28 Sep 2016, 23:14
2
27
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Difficulty:

55% (hard)

Question Stats:

57% (01:15) correct 43% (01:42) wrong based on 182 sessions

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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

Can someone please tell me why 4x4x4=64 is not the right answer?
There are 4 ways to fill the 1st box, and 4 ways to the 2nd box and 4 ways to fill the 3rd box..since it says that any box can contain any number of balls..i assumed that repetition is allowed.

Kindly explain.

Thanks alot!!!

Originally posted by geetgmat on 28 Sep 2016, 10:40.
Last edited by Bunuel on 28 Sep 2016, 23:14, edited 1 time in total.
Renamed the topic and edited the question.
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Posts: 4125
Re: In how many ways can we put 4 different balls in 3 different boxes whe  [#permalink]

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28 Sep 2016, 15:23
4
Top Contributor
1
geetgmat wrote:
In how many ways can we put 4 different balls in 3 different boxes when any box can contain any number of balls?

80
81
64
63
82

Can someone please tell me why 4x4x4=64 is not the right answer?
There are 4 ways to fill the 1st box, and 4 ways to the 2nd box and 4 ways to fill the 3rd box..since it says that any box can contain any number of balls..i assumed that repetition is allowed.

Kindly explain.

Thanks alot!!!

"There are 4 ways to fill the 1st box"
This assumes that the box needs to be filled.
No requirement for any one box to contain ANY balls.

On the other hand, every ball needs to be placed somewhere.
So, we need to examine the number of ways to place each ball. See my solution above.

Cheers,
Brent
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Re: In how many ways can we put 4 different balls in 3 different boxes whe  [#permalink]

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28 Sep 2016, 11:21
geetgmat wrote:
In how many ways can we put 4 different balls in 3 different boxes when any box can contain any number of balls?

80
81
64
63
82

Can someone please tell me why 4x4x4=64 is not the right answer?
There are 4 ways to fill the 1st box, and 4 ways to the 2nd box and 4 ways to fill the 3rd box..since it says that any box can contain any number of balls..i assumed that repetition is allowed.

Kindly explain.

Thanks alot!!!

NOTE : The number of permutations/arrangments of n things , taken r at ta time when each item may be repeated once, twice...up to r times in any arrangement is $$n^r$$ ways.

The first box can be filled in n ways, the second box can be filled in n ways(even though the first box is filled with one item, the same item can be used for filling the second box also because repetition is allowed), the third box can also be filled in n ways...

The rth box can be filled in n ways..

Now all the r boxes together can be filled in (n*n*n*....r times ) ie. $$n^r$$ ways.

So $$3^4$$ ways..

Hope it clears..
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Joined: 12 Sep 2015
Posts: 4125
Re: In how many ways can we put 4 different balls in 3 different boxes whe  [#permalink]

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28 Sep 2016, 15:20
1
Top Contributor
3
geetgmat wrote:
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

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)

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Re: In how many ways can we put 4 different balls in 3 different boxes whe  [#permalink]

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12 Mar 2017, 09:24
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, total number of possibilities is 3*3*3*3 = 81
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Posts: 945
Re: In how many ways can we put 4 different balls in 3 different boxes whe  [#permalink]

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30 Dec 2018, 12:40
GMATPrepNow wrote:
geetgmat wrote:
In how many ways can we put 4 different balls in 3 different boxes when any box can contain any number of balls?

80
81
64
63
82

Can someone please tell me why 4x4x4=64 is not the right answer?
There are 4 ways to fill the 1st box, and 4 ways to the 2nd box and 4 ways to fill the 3rd box..since it says that any box can contain any number of balls..i assumed that repetition is allowed.

Kindly explain.

Thanks alot!!!

"There are 4 ways to fill the 1st box"
This assumes that the box needs to be filled.
No requirement for any one box to contain ANY balls.

On the other hand, every ball needs to be placed somewhere.
So, we need to examine the number of ways to place each ball. See my solution above.

Cheers,
Brent

How could people not give Kudos for such valuable information! Members please appreciate valuable contribution such as this one by giving Kudos! Thanks GMATPrepNow.
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Re: In how many ways can we put 4 different balls in 3 different boxes whe  [#permalink]

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03 Mar 2019, 03:31
1
Another way to think about it by considering the number of balls in each box.

Situation 1
Box 1: 4 balls | 0 balls | 0 balls
Box 2: 0 balls | 4 balls | 0 balls
Box 3: 0 balls | 0 balls | 4 balls

So situation 1 has 3 different possibilities. Similarly, situation 2, 3, and 4 have 3 different possibilities each.

Situation 2
Box 1: 3 balls
Box 2: 1 balls
Box 3: 0 balls

Situation 3
Box 1: 2 balls
Box 2: 2 balls
Box 3: 0 balls

Situation 4
Box 1: 2 balls
Box 2: 1 balls
Box 3: 1 balls

Thus the total number of ways in which 4 different balls can be put in 3 different boxes when any box can contain any number of balls is
3 x 3 x 3 x 3 = 81
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In how many ways can we put 4 different balls in 3 different boxes whe  [#permalink]

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26 Aug 2019, 13:50
geetgmat wrote:
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

given: 4 dif balls, 3 dif boxes, any number per box;

$$4•4•4=81$$

$${400}={3!/2!}•4c4=3$$ ("/2!" because 00 are identical)
$${310}={3!}•4c3=6•4=24$$
$${220}={3!}•4c2/2!=6•3=18$$ ("/2!" because 22 is double counting)
$${211}={3!}•4c2=6•6=36$$
$${total}=3+24+18+36=81$$

In how many ways can we put 4 different balls in 3 different boxes whe   [#permalink] 26 Aug 2019, 13:50
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