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If 2 people have 2 days off per week each, what are the odds that they would have at least one day off that is the same?
How do you solve this? Any help?

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    $\begingroup$ What is the probability that all are different? $\endgroup$ – saulspatz May 9 '18 at 20:00
  • $\begingroup$ The probability of getting any combination of off days off is the same for the person. Ex.: (Monday/Tuesday's probability is the same as Tuesday/Sunday) $\endgroup$ – Filipe F. May 9 '18 at 20:04
  • $\begingroup$ For each person, are the two days off consecutive? $\endgroup$ – Joffan May 9 '18 at 20:05
  • $\begingroup$ No, they can be any combination of days $\endgroup$ – Filipe F. May 9 '18 at 20:11
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    $\begingroup$ Welcome to MSE! As a minimum you're expected to show what you have tried and where you are stuck. $\endgroup$ – samerivertwice May 9 '18 at 20:16
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There are $\binom{7}{2}^2$ ways they can choose their days off. Of these, $\binom{7}{2}\binom{5}{2}$ involve no overlap. Therefore, the probability of some overlap is $1-\dfrac{\binom{5}{2}}{\binom{7}{2}}=1-\dfrac{5\times 4}{7\times 6}=\dfrac{11}{21}$.

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  • $\begingroup$ @Joffan I advise you to make that a separate question. That'll give you room to decide what counts as adjacent. For example, are the first & last days of the week adjacent? $\endgroup$ – J.G. May 9 '18 at 20:05
  • $\begingroup$ I arrived to the same answer, but I was asking myself how to properly explain to my peers. I couldn't properly explain why you would use the (5|2) part? (sorry I can't format it properly) $\endgroup$ – Filipe F. May 9 '18 at 20:14
  • $\begingroup$ @FilipeF. Because if you try to list the no-overlap combinations, each time you specify one person's days off only $5$ days are available for the second person to choose from. $\endgroup$ – J.G. May 9 '18 at 20:18
  • $\begingroup$ @FilipeF. Welcome to MathSE. This tutorial explains how to typeset mathematics on this site. $\endgroup$ – N. F. Taussig May 10 '18 at 10:03

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