In how many ways can 7 identical erasers be distributed among 4 kids i

Question

In how many ways can 7 identical erasers be distributed among 4 kids in such a way that each kid gets at least one eraser but nobody gets more than 3 erasers?

A.13
B.14
C.15
D.16
E.20

KarishmaB · Answer

Each kid has to get at least one eraser so we give away 4 erasers to 4 kids. Now we are left with 3 erasers for 4 kids. They can be given away in 2 ways: Case1: 1 - 1 - 1 3 kids get one eraser each in 4C3 ways = 4 ways (select 3 kids of the 4 who will get erasers) Case 2: 2 - 1 One kid gets 2 erasers and another gets 1 in 4C1 * 3C1 = 12 ways In all we have 4 + 12 = 16 ways to distribute erasers. Answer (D) For discussion on Combinations, check out my video: https://www.youtube.com/watch?v=tUPJhcUxllQ

Archit3110 · Answer

total possible ways
ABCD ; 1,1,1,0 ; 4!/3! ; 4 ways
ABCD; 2,1,0,0;4!/2! ; 12 ways
total ; 16
IMO D

AyushCrackVerbal · Answer

Each kid must get at least 1 eraser and all the erasers are identical. So there is only one way on which we can distribute 4 erasers to 4 kids.

Now, we are left with 3 erasers that need to be distributed among 4 kids.

Since no student can get more than 3 erasers, we can not give the remaining 3 erasers to one student only.

Therefore, 2 cases are possible:

CASE 1

1 eraser each is given to 3 students:

No. of possible ways =  4C3  =  4

CASE 2

1 eraser is given to 1 student and 2 erasers are given to another student:

No. of possible ways =  4C1  *  3C1  = (4*3) =  12

Total number of ways =  12 + 4  = 16

So the correct option is  D

Kinshook · Answer

Asked: In how many ways can 7 identical erasers be distributed among 4 kids in such a way that each kid gets at least one eraser but nobody gets more than 3 erasers?

E |E |E |E

4 identical erasers are distributed among 4 kids and remaining 3 erasers are to be distributed among 4 kids

Number of ways = 6C3 = 20 ways
But no kid should get more than 3 erasers.
Therefore, the cases when remaining 3 erasers are distributed to 1 kids are to be excluded = 4 ways

Number of ways in which7 identical erasers can be distributed among 4 kids in such a way that each kid gets at least one eraser but nobody gets more than 3 erasers = 20 - 4 = 16 ways

IMO D

CrackverbalGMAT · Answer

There are 2 important equations for these types of arrangements

If the number of non negative integer solutions for the equation  x_1 + x_2 \space + \space ..+ \space x_n \space = \space n , then the number of ways the distribution can be done is:

(i)  ^{n+r−1}C_{r−1} . In this case, value of any variable can be zero.

(ii)  ^{n−1}C_{r−1} . In this case, minimum value for any variable is 1.

Given that: n = 7 erasers, r = 4 children and and each child should have at least one eraser, but the maximum should not exceed 3.

Total number of ways, where each child gets at least 1 eraser =  ^{n-1}C_{r-1} = \space ^{7-1}C_{4-1} = \space ^6C_3 = \frac{6 \space * \space 5 \space * \space 4}{3 \space * \space 2 \space * \space 1} = 20

These 20 also include the cases, where 1 child gets all 4 erasers. The total such cases is 4.

Therefore the required number of cases = 20 - 4 = 16

Option D

Arun Kumar

sayan640 · Answer

CrackverbalGMAT  How do you know that total such cases is 4 ?   KarishmaB   MartyMurray  Can you please enlighten ?

sayan640 · Answer

I think total such cases when a child gets more than 3 is 4 because after you distribute 4 identical chocolates among 4 children , you are left with 3 chooclates. The only way a child can get more than 3 chocolates is when these remaining 3 chocolates go into one particular child . There are 4 such children . Hence , there can be 4 ways in which a child can get more than 3 chocolates.
Say , there are 4 children such as A, B , C and D. 
After each got 1 chocolate , either A can get all the remaining three ( thus making his count 1 + 3 = 4 which is more than 3 ) or B can get all the remaning three or C or D. Hence 4 such cases are possible.
 KarishmaB  , kindly check my reasoning.

Regor60 · Answer

This is a correct statement.

Posted from my mobile device

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

In how many ways can 7 identical erasers be distributed among 4 kids i

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

In how many ways can 7 identical erasers be distributed among 4 kids i

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards