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Combinatorics

Counting is a bit harder than 1, 2, 3, ...

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How many 5-digit integers have the property that every pair of adjacent digits have different parity?

Parity means oddness/evenness, so 12345 satisfies the property but 12435 does not, since 2 and 4 have the same parity.

A two-digit number is randomly chosen. What is the probability that the product of the digits is a multiple of 3?

The New Jersey Tigers and the Massachussetts Beavers are playing for a basketball championship. They play a best of 7 game series. (They keep playing games against each other until one team has won 4 games. There are no ties.) If the series lasts 7 games with the Tigers winning, how many different ways could the outcomes of the games have gone?

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