# 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?

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?

• What is the probability that all are different? – saulspatz May 9 '18 at 20:00
• 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) – Filipe F. May 9 '18 at 20:04
• For each person, are the two days off consecutive? – Joffan May 9 '18 at 20:05
• No, they can be any combination of days – Filipe F. May 9 '18 at 20:11
• Welcome to MSE! As a minimum you're expected to show what you have tried and where you are stuck. – samerivertwice May 9 '18 at 20:16

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 $1-\dfrac{\binom{5}{2}}{\binom{7}{2}}=1-\dfrac{5\times 4}{7\times 6}=\dfrac{11}{21}$.
• @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. – J.G. May 9 '18 at 20:18