Question

Three students are chosen at random from a school. If the birthdays of the three students fall in the same week, what is the probability that their birthdays are not on three different days of the week?

• A. $\frac{15}{49}$
• B. $\frac{30}{49}$
• C. $\frac{6}{49}$
• D. $\frac{19}{49}$

Answer 1

The correct option is D $\frac{19}{49}$

Let us calculate the probability that their birthdays fall on three different weekdays.
All the three persons can have their birthdays on any day out of the 7 days. So, the number of possible combinations of the days on which their birthdays might fall $=7\times 7\times 7=343$.
But according to the question, the birthday of the first person can be on any day out of 7 days, the birthday of the second person can be on any day out of any 6 days and the birthday of the third person can be on any day out of 5 days.
So, the combinations with the favorable outcomes $=7\times 6\times 5=210$
Probability that their birthdays are on three different days $=\frac{\text{number of favorable outcomes}}{\text{total number of outcomes}}=\frac{210}{343}=\frac{30}{49}$
Now probability that their birthdays are not on three different days $=1-\frac{30}{49}=\frac{19}{49}$