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

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. 14
B. 15
C. 16
D. 17
E. 20

ScottTargetTestPrep · Accepted Answer

We can see that each child gets a positive integer number of erasers. So let’s first list the ways 4 positive integers can sum to 7 (assuming any integer in the list can’t be greater than the previous one).

3 + 2 + 1 + 1 = 7 and 2 + 2 + 2 + 1 = 7

So we have two sets {3, 2, 1, 1} and {2, 2, 2, 1}. However, take the set {3, 2, 1, 1}, for example, and let’s say the four children are A, B, C, and D. A having 3, B having 2, C having 1 and D having 1 eraser is different from A having 1, B having 3, C having 2 and D having 1 eraser. So we need to ask ourselves: How many ways can we permute the four numbers in each set? The answer is to use the formula for permutation of indistinguishable items. Since we have:

4!/2! = 24/2 = 12 ways to permute the 4 numbers in {3, 2, 1, 1} and

4!/3! = 24/6 = 4 ways to permute the 4 numbers in {2, 2, 2, 1},

there are a total of 12 + 4 = 16 ways to distribute 7 erasers among 4 children in such a way that each child gets at least one eraser but no child gets more than 3 erasers.
 
 Answer: C

nick1816 · Answer

x_1+x_2+x_3+x_4 = 7

Number of positive solutions = (3+4-1)C (3-1) = 6C3=20

Number of ways in which 1 person will get 4 erasers in such a way that each kid gets at least one eraser= 4C1=4

Number of ways 7 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

suchita2409 · Answer

Erasers can be distributed in 2 ways only.

1:
(2 2 2 1) this can be arranged in 4 ways.
2:
(3 2 1 1) this can be arranged in 4 ways as well 
4 ×4 =16 ways

Posted from my mobile device

nick1816 · Answer

(3 2 1 1) can be arranged in 4!/2! = 12 ways

Also both cases are not independent, so you can't multiply.

sayan640 · Answer

KarishmaB   MartyMurray  Cant we solve this question using the formulae for identical objects i.e (n-1 ) c (r-1) ?

malfoy · Answer

If there are 7 erasers to be distributed amongst 4 kids and each one at least gets 1, we can look at the question this way:
3 (7-4) erasers are to be distributed amongst 4 kids. Why? if each kid gets one instead of 7 we have 3 left to be distributed.

Now the challenge is no one can gets all 3 (even if someone gets 2 the number of erasers would be max 3) as question says we can't give out more than 3 erasers.
Using (n+r-1)c(r-1) we will have 6c3 = 20 {Distributing 3 erasers amongst 4 kids without any limits}
From 20 ways we need to subtract number of ways where all 3 erasers go to one kid i.e 4 = 16 which is the answer

However, I find Scott's solution much simpler, this is just using Stars and Bars for the sake of using it.

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

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

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