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dabaobao
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In how many ways can 5 apples (identical) be distributed among 4 ident 22 May 2019, 16:01
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A
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95% (hard)

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34% (01:36) correct 66% (01:20) wrong based on 296 sessions

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In how many ways can 5 apples (identical) be distributed among 4 identical baskets?

A) 6
B) 20
C) 51
D) 56
E) 4^5
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A
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In how many ways can 5 apples (identical) be distributed among 4 ident 22 May 2019, 16:02
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In how many ways can 5 apples (identical) be distributed among 4 identical baskets?

A) 6
B) 20
C) 51
D) 56
E) 4^5
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Official Solution


Credit: Veritas Prep


As we have seen previously, 5 fruits can be split into 4 groups in the following ways: {5, 0, 0, 0}, {4, 1, 0, 0}, {3, 2, 0, 0}, {3, 1, 1, 0}, {2, 2, 1, 0}, {2, 1, 1, 1}

Here, {5, 0, 0, 0} means that one basket has all 5 apples and the rest of the baskets are empty. Since the baskets are all identical, there is only 1 way of doing this.
{4, 1, 0, 0} means that one basket has 4 apples, another one has 1 apple and the rest of the baskets are empty. Since the fruits and the baskets are all identical, there is again only 1 way of doing this.

You have to select neither the fruits nor the baskets since they are all identical. You only have to decide how to distribute the apples in 4 groups. Therefore, each one of these cases will give us only one different way of distributing the fruits. Since there are 6 such cases, there are only 6 different ways of distributing the fruits.

ANSWER: A
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In how many ways can 5 apples (identical) be distributed among 4 ident Updated on: 11 Dec 2019, 10:49
Originally posted by exc4libur on 11 Dec 2019, 09:30.
Last edited by exc4libur on 11 Dec 2019, 10:49, edited 1 time in total.
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dabaobao
In how many ways can 5 apples (identical) be distributed among 4 identical baskets?

A) 6
B) 20
C) 51
D) 56
E) 4^5
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Total partitions = 6
5000
4100
3200
3110
2111
2210

Ans (A)
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In how many ways can 5 apples (identical) be distributed among 4 ident 11 Dec 2019, 10:36
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exc4libur
dabaobao
In how many ways can 5 apples (identical) be distributed among 4 identical baskets?

A) 6
B) 20
C) 51
D) 56
E) 4^5

Total partitions = 6
5000
4100
3200
3110
2111
2110 It Should be 2210

Ans (A)
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Dear exc4libur ,

I think u made one typo mistake in your answer. I have highlighted it.
Please correct.

Regards,
Rajat Chopra
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In how many ways can 5 apples (identical) be distributed among 4 ident 12 Dec 2019, 19:49
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dabaobao
In how many ways can 5 apples (identical) be distributed among 4 identical baskets?

A) 6
B) 20
C) 51
D) 56
E) 4^5
Show more

The 5 identical apples can be distributed among 4 identical baskets in the following ways:

{5, 0, 0, 0}

{4, 1, 0, 0}

{3, 2, 0, 0}

{3, 1, 1, 0}

{2, 2, 1, 0}

and

{2, 1, 1, 1}

We see that there are 6 ways.

(Note: Since the apples are identical and the baskets are also identical, a rearrangement such as {0, 1, 0, 4} is considered the same as {4, 1, 0, 0}.)

Answer: A
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In how many ways can 5 apples (identical) be distributed among 4 ident 14 Dec 2019, 21:48
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Is here a mathematical formula to solve this question?
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In how many ways can 5 apples (identical) be distributed among 4 ident 19 Sep 2020, 11:45
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[quote="dabaobao"]In how many ways can 5 apples (identical) be distributed among 4 identical baskets?

Number of ways = {(5,0,0,0),(4,1,0,0),(3,1,1,0),(2,1,1,1),(2,2,1,0),(3,2,0,0)} = 6

IMO A
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In how many ways can 5 apples (identical) be distributed among 4 ident 22 Feb 2024, 22:21
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dabaobao
In how many ways can 5 apples (identical) be distributed among 4 identical baskets?

A) 6
B) 20
C) 51
D) 56
E) 4^5
Show more
­I thought of using the formula \((n+r-1)C(r-1)\) but here both apples and baskets are identical so we cannot use this. Here are the possible distributions.
(5000), (1400), (1130), (1112), (1220) and (2300). There will be no rearrangements since baskets are identical so only 6 ways are possible. Option (A) is correct.
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In how many ways can 5 apples (identical) be distributed among 4 ident 23 Apr 2025, 01:45
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