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

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Math Expert
Joined: 02 Sep 2009
Posts: 64203
In how many ways can 7 identical erasers be distributed among 4 kids  [#permalink]

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02 Apr 2020, 09:59
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Difficulty:

45% (medium)

Question Stats:

43% (01:23) correct 57% (02:11) wrong based on 29 sessions

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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

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Re: In how many ways can 7 identical erasers be distributed among 4 kids  [#permalink]

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02 Apr 2020, 14:47
1
$$x_1+x_2+x_3+x_4 = 7$$

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

Bunuel wrote:
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
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Joined: 11 May 2019
Posts: 42
Re: In how many ways can 7 identical erasers be distributed among 4 kids  [#permalink]

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02 Apr 2020, 15:35
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

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Re: In how many ways can 7 identical erasers be distributed among 4 kids  [#permalink]

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02 Apr 2020, 17:45
1
(3 2 1 1) can be arranged in 4!/2! = 12 ways

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

suchita2409 wrote:
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

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Re: In how many ways can 7 identical erasers be distributed among 4 kids  [#permalink]

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04 Apr 2020, 13:31
Bunuel wrote:
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

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.

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Re: In how many ways can 7 identical erasers be distributed among 4 kids   [#permalink] 04 Apr 2020, 13:31