[BrianL]

Originally Posted by Velvet

A music festival award gives awards for 1st, 2nd and 3rd place in each category. If there are 7 contestants for piano and 9 for violin, in how many different ways could the 6 awards be presented?

There are a total of 6 awards, assuming per your problem statement the awards can be presented in any order, then the answer is 6! (6 factorial) or 720. This is the number of ways 6 distinct things can be orderd.

[bopat]

It's not clear from the question how many categories there are or how many awards are given in each category. In order to answer the question as stated, we would need more information. If there are two categories, piano and violin, and each category has three awards (1st, 2nd, and 3rd), then there are 7 * 3 = <<73=21>>21 different ways to award the piano prizes and 9 * 3 = <<93=27>>27 different ways to award the violin prizes, for a total of 21 * 27 = <<21*27=567>>567 ways to award all of the prizes.

[JMintzer]

[tuccillo]

There are two categories and 3 awards in each category. See posts #4, #7, and #10. This is, essentially, an exercise in computing permutations.

Originally Posted by bopat

It's not clear from the question how many categories there are or how many awards are given in each category. In order to answer the question as stated, we would need more information. If there are two categories, piano and violin, and each category has three awards (1st, 2nd, and 3rd), then there are 7 * 3 = <<73=21>>21 different ways to award the piano prizes and 9 * 3 = <<93=27>>27 different ways to award the violin prizes, for a total of 21 * 27 = <<21*27=567>>567 ways to award all of the prizes.