PUNE MCA Exam 2016 Question Paper with answers for online practice

Concept:The die has outcomes 0 (blank), 2, and 3 with probabilities $\frac{4}{6}$, $\frac{1}{6}$, and $\frac{1}{6}$ respectively. The sum of five throws must be 12.Explanation:Let $x$ be the number of 2's, $y$ the number of 3's, and $z$ the number of 0's. Then $x + y + z = 5$ and $2x + 3y = 12$.Solving gives two valid cases: $(x=3,y=2,z=0)$ and $(x=0,y=4,z=1)$.Total possible outcomes (considering each face equally likely): $6^5 = 7776$.Case 1 (three 2's, two 3's): Choose positions for the 2's: $\binom{5}{3}=10$ ways. Each 2 and 3 come from a single face, so $10$ sequences.Case 2 (four 3's, one 0): Choose position for the 0: $\binom{5}{1}=5$ ways. The 0 can be any of the 4 blank faces, so $5 \times 4 = 20$ sequences.Total favorable sequences: $10 + 20 = 30$.Probability $= \frac{30}{7776} = \frac{5}{1296}$.Answer:The probability $\frac{5}{1296}$ is not listed among the options.