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In how many ways can Nicole distribute 20 chocolates among B 11 Dec 2013, 02:02
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51% (01:00) correct 49% (01:20) wrong based on 489 sessions

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In how many ways can Nicole distribute 20 chocolates among Ben, Kate, Mark and Tom so that each receives at least one?

A. 20C4
B. 19C3
C. 21C4
D. 21C3
E. None of the answers is correct
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Bunuel
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In how many ways can Nicole distribute 20 chocolates among B 11 Dec 2013, 03:33
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In how many ways can Nicole distribute 20 chocolates among Ben, Kate, Mark and Tom so that each receives at least one?

A. 20C4
B. 19C3
C. 21C4
D. 21C3
E. None of the answers is correct
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The total number of ways of dividing n identical items among r persons, each one of whom receives at least one item is \(n-1C_{r-1}\).

Thus the answer is 19C3.

Answer: B.

P.S. Does not seem that this question is from reliable GMAT source...
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In how many ways can Nicole distribute 20 chocolates among B 11 Dec 2013, 03:38
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In how many ways can Nicole distribute 20 chocolates among Ben, Kate, Mark and Tom so that each receives at least one?

A. 20C4
B. 19C3
C. 21C4
D. 21C3
E. None of the answers is correct

The total number of ways of dividing n identical items among r persons, each one of whom receives at least one item is \(n-1C_{r-1}\).

Thus the answer is 19C3.

Answer: B.

P.S. Does not seem that this question is from reliable GMAT source...
Show more

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Hope this helps.
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In how many ways can Nicole distribute 20 chocolates among B 15 Feb 2024, 00:18
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fahdman
Hello Bunuel,

Please why 20c4 does not work here? aren't we sure that at least each one will get a chocolate? can you explain the answer with a different method because I can't retain formulas. Thank you
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­In how many ways can Nicole distribute 20 chocolates among Ben, Kate, Mark and Tom so that each receives at least one?

A. 20C4
B. 19C3
C. 21C4
D. 21C3
E. None of the answers is correct

When distributing identical objects, we should use the "Stars and Bars" method. 

"Stars and Bars" method:

Since each Ben, Kate, Mark and Tom must get at least 1 chocolate, we are essentially distributing only the remaining 20 - 4 = 16 chocolates.

Imagine the 16 chocolates as 16 stars in a row. To divide these stars among 4 people, we need 3 bars to create 4 separate sections. For example:

    ******|****|***|*** would represent that the first one got 6 chocolates, the second got 4, the third got 3, and the fourth got 3.

    ||**********|****** would represent that the first one got 0 chocolates, the second got 0, the third got 10, and the fourth got 6.

    *******||******|*** would represent that the first one got 7 chocolates, the second got 0, the third got 6, and the fourth got 3.

So, the problem becomes one of arranging these 16 identical stars and 3 identical bars in a row, which is given by 19C3 = 19!/(16!3!) = 969.

Answer: B

Hope it helps.­

P.S. Check other questions related to distribuition in DISTRIBUTING ITEMS/PEOPLE/NUMBERS... (QUESTION COLLECTION) from our Special Questions' Directory.­
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In how many ways can Nicole distribute 20 chocolates among B 11 Dec 2013, 05:07
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The source is "Ultimate guide to GMAT math" by Jay Stewart

I got the same answer in the book, what interests me is the logic behind this formula. thx!
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In how many ways can Nicole distribute 20 chocolates among B 14 Feb 2024, 17:47
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Hello Bunuel,

Please why 20c4 does not work here? aren't we sure that at least each one will get a chocolate? can you explain the answer with a different method because I can't retain formulas. Thank you
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In how many ways can Nicole distribute 20 chocolates among B 25 Oct 2025, 23:55
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