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1) in how many ways one can enter hexagonal room having door on each wall that he exits from some different door?

2)how many tricolor flag can be designed from the rainbow color?

1) For choosing an entrance we have $\binom{6}{1}$ choices, and for corresponding exits, we have$\binom{5}{1}$ choices.

So total ways = 6 * 5 = 30

2) Rainbow has 7 colors. We need 3 colors to form a tricolor flag.

So total tricolor flags possible = $\binom{7}{3}$ * 3! = 210

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