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Seek simple and succinct solutions in these systems by sussing-out the significant symmetries.

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Paddy flips a fair coin \(6\) times, and writes down each result in order. What is the total number of possible outcomes?

Note: The outcome \(THHH\) is different from the outcome \(HHHT\).

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How many ways are there to color the above regions with 4 different colors, if adjacent regions should have different colors?

**Note:** You need not use all four colors but each region must be colored by any one of them.

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How many odd 3-digit numbers are there, whose digits are distinct integers from the set \( \{0, 1, 2, 3, 4, 5, 6, 7 \} \).

This problem is posed by Gabriel M.

**Details and assumptions**

The number \(12=012\) is a 2-digit number, not a 3-digit number.

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