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Question :If there are $6$ periods in each working day of a school,in how many can one arrange $5$ subjects such that each subject is allowed at least one period?

My solution: ${^5P_5} *{^5P_1}=600$

Logic that I used is:out of 6 periods,5 periods are arranged for $5$ subjects and the remaining one period is arranged for any of the $5$ subjects.

But my solution does not match with the one provided by textbook.Where is my mistake?

And is not it a combination's problem (since the order of the arrangement of classes does not matter here)?

Graham Kemp
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  • Your solution is counting duplicate arrangements. – barak manos May 16 '15 at 14:44
  • Could you please explain? –  May 16 '15 at 14:46
  • Could you please explain the obscure statement "each subject is allowed at least one period"? – barak manos May 16 '15 at 14:47
  • You are in school and there are periods such that in each period a particular subject is taught....here –  May 16 '15 at 14:51
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    One of the five subjects must appear twice, and you state the Question as a matter of "how many ways can one arrange" the subjects. So I don't follow why "the order of the arrangement of classes does not matter here". – hardmath May 16 '15 at 14:51

1 Answers1

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Note: The order of the subject matters since we are counting the ways to arrange the subjects.

Now since each subject must have at least one period, then clearly, unless a free period is allowed, one of the subjects will have two periods.

You must count the ways to:

  • choose which is that doubled subject
  • the ways to choose its two periods (they don't have to be adjacent), and then
  • the ways to arrange four singletons among the remaining periods.

Multiply these three counts together.

Graham Kemp
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