UPSC NDA 2 2017 Mathematics Solved Paper

Concept:The probability that at least two persons share a birth month is found by adding the probability that all three share a month and the probability that exactly two share a month.Explanation:Assume each month is equally likely as a birth month.Total number of possible birth-month combinations (ordered) for three persons = $12^3$.Case 1: All three born in the same monthNumber of favourable ways = $12$ (any one of the 12 months).Probability = $\frac{12}{12^3} = \frac{1}{144}$.Case 2: Exactly two born in the same monthChoose which two persons share a month: $\binom{3}{2} = 3$ ways.Choose the month for the pair: $12$ ways.Choose a different month for the third person: $11$ ways.Favourable ways = $3 \times 12 \times 11$.Probability = $\frac{3 \times 12 \times 11}{12^3} = \frac{33}{144}$.Total probability (at least two same month)Add the two probabilities: $\frac{1}{144} + \frac{33}{144} = \frac{34}{144} = \frac{17}{72}$.Answer:$\boxed{\frac{17}{72}}$ (Option B).