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In How Many Ways Can Chocolates Be Distributed To People Such That No One Gets More Than ?

The document discusses how to distribute 10 chocolates to 3 people such that no person receives more than 5 chocolates. It suggests using the inclusion-exclusion principle to calculate the number of ways by considering the events that each person receives at least 5 chocolates, and subtracting intersections of these events since it is impossible for all 3 people to receive at least 5 chocolates given there are only 10 chocolates total. Another user confirms this approach, writing out that the number of ways is P(A1) + P(A2) + P(A3) - P(A1∩A2) - P(A1∩A3) - P(A2∩A

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Echizen Ryoma
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55 views1 page

In How Many Ways Can Chocolates Be Distributed To People Such That No One Gets More Than ?

The document discusses how to distribute 10 chocolates to 3 people such that no person receives more than 5 chocolates. It suggests using the inclusion-exclusion principle to calculate the number of ways by considering the events that each person receives at least 5 chocolates, and subtracting intersections of these events since it is impossible for all 3 people to receive at least 5 chocolates given there are only 10 chocolates total. Another user confirms this approach, writing out that the number of ways is P(A1) + P(A2) + P(A3) - P(A1∩A2) - P(A1∩A3) - P(A2∩A

Uploaded by

Echizen Ryoma
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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combinatorics - In how many ways can $10$ chocolates be distributed to... https://math.stackexchange.com/questions/1709873/in-how-many-ways-...

In how many ways can chocolates be distributed to


people such that no one gets more than ?
Asked 3 years, 5 months ago Active 3 years, 5 months ago Viewed 1k times

chocolates are distributed to people such that no one gets more than or . How
many ways can this be done?
1 I know how to do this if the condition is every person gets at least one. Here the condition is
different. So need help.

combinatorics

1
edited Mar 23 '16 at 10:00 asked Mar 23 '16 at 7:28
N. F. Taussig Devedrakumar V Shah
48.8k 10 36 60 50 4

chocolates? – Crostul Mar 23 '16 at 7:36

Zero chocolates are distributed to three persons? There is clearly only one way for that to occur, namely
when everyone gets none. If you had a number larger than , then approach via inclusion-exclusion on
whether or not the upper bound requirement for each of is violated. – JMoravitz Mar 23 '16 at 7:37

Problem needs to be edited. There is only one way to distribute chocolates. ... Also, you need to explain
what tools you have available. Inclusion/Exclusion? Generating functions? And what have you tried?
– Christopher Carl Heckman Mar 23 '16 at 7:37

Sorry it is not 0 chocolates. It is 10 chocolates. My mistake in typing – Devedrakumar V Shah Mar 23 '16
at 7:44

The end result is the same as what I mentioned already. Let denote the event that the upper bound
condition is violated for person , in other words . In other words . You say you know how to
do it if everyone gets at least one, you should then also know how to do it if a specific person gets at least
five, or multiple people get at least five. – JMoravitz Mar 23 '16 at 14:43

Apply inclusion-exclusion. We try to find

noting that in this case intersections of all three events is empty (since there are only ten chocolates, it is
impossible for all three people to have at least five). – JMoravitz Mar 23 '16 at 14:45

2 Answers

Set:

1 of 1 8/27/2019, 11:42 AM

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