Probability
Probability
Allen Star Batch
Grade 12
Question:
Six fair dice are thrown independently. The probability that there are exactly 2 different pairs (A pair is an ordered combination like 2, 2, 1, 3, 5, 6) is $p$, then $4p$ is $\ldots\ldots\ldots\ldots$
Step-by-Step Solution
Key Concept: Use combinations to select numbers, then apply permutations with repetition formula to arrange them.
Total outcomes when rolling a die 6 times is $6^6$. We need to choose 4 different numbers from 6: $\binom{6}{4}$ ways. Then choose 2 of the remaining 4 numbers to appear twice: $\binom{4}{2}$ ways. Arrange 6 numbers where 2 pairs are identical: $\frac{6!}{2!2!} = 180$ ways. Thus probability $= \frac{\binom{6}{4} \times \binom{4}{2} \times 180}{6^6} = \frac{15 \times 6 \times 180}{46656} = \frac{26}{72}$.
Correct Answer: 1.44