Five balls of different colors are to be placed in three boxes of diff

Question

Five balls of different colors are to be placed in three boxes of different sizes. Each box can hold all five. In how many different ways can we place the balls so that no box remains empty?

A. 100
B. 125
C. 150
D. 175
E. 200

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GMATinsight · Accepted Answer

Case 1: {1, 1, 3} two boxes with one ball and one box with 3 balls Select box which gets three balls = ^3C_1 = 3 Select 3 balls for that box = ^5C_3 = 10 Remaining two balls to be put into two boxes in 2! = 2 ways Desired outcomes = 3*10*2 = 60 ways Case 2: {1, 2, 2} two boxes with two balls each and one box with 1 balls Select boxes which gets two balls each = ^3C_2 = 3 Select 2 balls for first of those selected box = ^5C_2 = 10 Select 2 balls for Second of those selected box = ^3C_2 = 3 Remaining one balls to be put into last box = 1 ways Desired outcomes = 3*10*3*1 = 90 ways My Answer = 60+90 = 150

sujoykrdatta · Answer

Possible distributions are:
3-1-1: 
Choose the balls: 5c3 x 2c1 x 1c1 = 20
Decide the arrangements: 3!/2! = 3
Total = 20*3 = 60
Or
2-2-1
Choose the balls: 5c2 x 3c2 x 1c1 = 30
Decide the arrangements: 3!/2! = 3
Total = 30*3 = 90

Total = 60+90 = 150

Alternatively:
Each ball has 3 options (boxes)
Total ways = 3^5 = 243

Cases to be removed:
1. Cases where all balls are in 1 box = 3
2. Cases where all balls are in 2 boxes:
Choose 2 boxes = 3c2 = 3
Each ball has 2 options = 2^5 = 32
But, of these, there are 2 cases when the balls are in only 1 box; these we should ignore as we considered this in 'Case 1'; ie. 32-2 = 30

So total = 3c2 x (32-2) = 90

Hence, required answer 
= 243 - 3 - 90 = 150

Bunuel  - Please check the options once

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ScottTargetTestPrep · Answer

Solution:

Since each box must have at least one ball, we see that there are only two cases to consider:

Case 1:  (3, 1, 1) or (1, 3, 1) or (1, 1, 3)

For (3, 1, 1), we have 5C3 x 2C1 x 1C1 = 10 x 2 x 1 = 20 ways to choose the balls. 
For (1, 3, 1), we have 5C1 x 4C3 x 1C1 = 5 x 4 x 1 = 20 ways to choose the balls.
For (1, 1, 3), we have 5C1 x 4C1 x 3C3 = 5 x 4 x 1 = 20 ways to choose the balls.

Case 1 yields a total of 20 + 20 + 20 = 60 ways.

Case 2:  (1, 2, 2) or (2, 1, 2,) or (2, 2, 1)

For (1, 2, 2), we have 5C1 x 4C2 x 2C2 = 5 x 6 x 1 = 30 ways to choose the balls.
Note that the two remaining choices (2, 1, 2,) or (2, 2, 1) yield 30 ways and 30 ways, respectively.

Thus, Case 2 yields 30 + 30 + 30 = 90 ways.

The total number of ways to place the balls such that no box remains empty is 60 + 90 = 150.

Answer: C

Fdambro294 · Answer

We are distributing 5 different balls into 3 different bins.

In this scenario, which ball is selected and the ordering of which bin it enters both matter.

Since each bin must receive at least 1 ball (“no bin can be empty”), there are only 2 scenarios in which the balls can be distributed:

Or

Scenario 1: first distribute the 5 different balls into stacks of 3 - 1 - 1

No. of ways this can be done is:

5! / (3! * 1! * 1!)

Because 2 of the stacks will have identical amounts of balls, each unique division will be over counted by 2.

5! / (3! * 1! * 1!) * (1 / 2!)

Now, we need to find the no. of ways we can arrange or shuffle around the 3 stacks such that they are placed in the 3 different bins.

Again, since all the balls are different and the bins are different, the no. of different arrangements is 3!

5! / (3! * 1! * 1!) * (1 / 2!) * (3!) =

60 ways

OR

Scenario 2:

Following the same logic as above, remembering that we again have 2 stacks of identical amounts of 2 and 2:

No. of ways = 5! / (2! * 2! * 1!) * (1 / 2!) * (3!) =

90 ways

60 + 90 = 150 ways

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Saumyojit · Answer

See where am i wrng?

For (3,1,1)

Choosing 3 out of 5 balls--> 5c3
Choosing 1 out of 2 balls --> 2c1
Choosing 1 out of 1 --> 1c1

Now for a particular combination (b1 b2 b3 , b4, b5)

Main point: For a single event suppose, Triplets have three options of sitting in any one box so it chooses any one Box , then there are two boxes left for 2 Single Distinct balls ; so ball 4 have 2 options so suppose b4 sits in any ONE out of 2 box and the remaining one box can be taken by b5 . So , 6 ways ?

So, 5c3 * 2c1 * 1c1 *  3!  .

What is the mistake that am i not able to catch?

GMATinsight · Answer

Saumyojit

For (3,1,1)

Choosing 3 out of 5 balls--> 5c3
Choosing 1 out of 2 balls --> 2c1
Choosing 1 out of 1 --> 1c1

Here you need to also identify the box which is getting 3 balls because it can be in these three ways (311)(131)(113)
You can calculate it in   ^3C_1   ways and  not in 3! ways 
so the calculation of this case will be =  ^5C_3* ^2C_1*^ 1C_1*   ^3C_1   = 60

Saumyojit · Answer

Your reply shows that you did not understand what my doubt is .

For a single event suppose, Triplets have three options of sitting in any one box so it chooses any one Box , then there are two boxes left for 2 Single Distinct balls ; so ball 4 have 2 options so suppose b4 sits in any ONE out of 2 box and the remaining one box can be taken by b5 . So , 6 ways .

Now what you are saying is that Triplets have three options of sitting in any one box so it chooses any one Box --> 3 ways . 
But what about the single ball (B4 , B5)

I identified the box which is getting 3 balls . What are u talking about.

Read the above part again

Saumyojit · Answer

box1 box2 box3

b1b2b3 , b4 , b5

b1b2b3 , b5 , b4

are these two different way or one way?

Saumyojit · Answer

Concept that was used first : Division of Objects into equal and unequal groups
This is where I was mistaking.

5c3 --> B1 B2 B3 chosen out of 10 combo
2c1--> b4 chosen out of 2 combo (b4 or b5)
1c1---> B5
Here i was dividing 5 balls in 3 groups **not** boxes in 3:1:1
Now, group 2 and group 3 is the same thing only .
(B1B2B3) , B4 ,B5 and (B1B2B3) , B5 ,B4 --> These two are same . SO, 2c1 is overcounting .
so, final expression of Division into Groups = (5c3 * 2c1 * 1c1 )/ 2!
So, Now comes the distribution part -->

For a event suppose, Triplets have three options of sitting in any one box so it chooses one , then there are two boxes left for 2 Single Distinct balls ; so ball 4 have 2 options so suppose b4 sits in any ONE out of 2 box and the remaining one box can be taken by b5 . So , 6 ways . 
3options * 2options * 1 option
triplets single another single 
So, Final expression of (3,1,1) = (5c3 * 2c1 * 1c1 ) * 3

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