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Counting problem with three types of objects

Source: RMO Maharashtra and Goa 2019 P5

November 10, 2019
combinatorics

Problem Statement

There is a pack of 27 distinct cards, and each card has three values on it. The first value is a shape from {Δ,□,⊙}\{\Delta,\square,\odot\}; the second value is a letter from {A,B,C}\{A,B,C\}; and the third value is a number from {1,2,3}\{1,2,3\}. In how many ways can we choose an unordered set of 3 cards from the pack, so that no two of the chosen cards have two matching values. For example we can chose {ΔA1,ΔB2,⊙C3}\{\Delta A1,\Delta B2,\odot C3\} But we cannot choose {ΔA1,□B2,ΔC1}\{\Delta A1,\square B2,\Delta C1\}
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