On a particular Sunday, a total of 53 customers enter a 24-hour laundromat to do their laundry where there are 10 washing machines available. If each customer uses only one washing machine and each washing machine is used by at most 7 customers, show that at least 6 washing machines are used by at least 4 customers.

[HINT: To use the Pigeonhole Principle for this problem, convert it into a problem with the “at least 1 pigeonhole” situation.

asked Oct 27, 2016 in Other Math Topics by Hana

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2 Answers

I don't know the answer to this question. Do you have perhaps like a graph or something to show what your information is?
answered Oct 27, 2016 by Anthonie Saunders Level 2 User (1,360 points)

Okay I know that I answered this 7 hours ago and it wasn't helpful but I seriously don't understand your question. If you could put more information in your question, perhaps I could solve it but for now I must leave it blank and provide you no answer due to your lack of information.

I hope I interpreted this question correctly. Here is my proposed proof.

Consider 24 hours as being the pigeon holes. When 25 customers have arrived at least 2 will have arrived in the same hour. When 49 customers arrive at least 3 will arrive in the same hour. So for 53 customers there will be 5 hours in which at least 3 customers arrive: 19*2+5*3=53.

Each machine does no more than 7 washes in 24 hours. So each machine must take at least about 3 hours to complete a wash (and dry).

In 3 hours, a complete wash cycle, at least 6 customers will have arrived. They will each use an available machine. So 6 machines at least will be in use.

Over the course of the wash cycle at least 6 more customers arrive, but there are only 4 machines available. While the new customers are completing their wash cycles the previous customers will have completed their wash cycles and the machines will become available for more new customers. Until they arrive there will be at least 4 customers in the laundrette.

answered Oct 29, 2016 by Rod Top Rated User (487,620 points)

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