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Olympiad workbook
Exam
Olympiad workbook
Subject
IOQM
Chapter
Permutation and Combination

Olympiad workbook · IOQM · Permutation and Combination

Unconventional dice are to be designed such that the six faces are...

Unconventional dice are to be designed such that the six faces are marked with numbers from 1 to 6 with 1 and 2 appearing on opposite faces. Further, each face is colored either red or yellow with opposite faces always of the same color. Two dice are considered to have the same design if one of them can be rotated to obtain a dice that has the same numbers and colors on the corresponding faces as the other one. Find the number of distinct dice that can be designedUnconventional dice are to be designed such that the six faces are marked with numbers from 1 to 6 with 1 and 2

Correct answer

24

Step-by-step solution

Fix 1 and 2 anywhere in 1 way Fix 3 anywhere in 1 way Unconventional dice are to be designed such that the six faces are marked with numbers from 1 to 6 with 1 and 2
Now remaining 3 faces can be filled in 3!2\frac{3!}{2} ways and coloring them in 2×2×22 \times 2 \times 2 ways ∴\therefore Total distinct dice =3!2×8=24=\frac{3!}{2} \times 8=24 ways
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