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Math Review Question: 10 Page: 297 Difficulty: medium

(a) In how many different ways can the judges award the 3 prizes?

Take the task of awarding prizes and break it into stages.

Stage 1: Select a contestant to receive the FIRST prize There are 8 contestants to choose from. So, we can complete stage 1 in 8 ways

Stage 2: Select a contestant to receive the SECOND prize There are 7 contestants REMAINING to choose from. So, we can complete stage 2 in 7 ways

Stage 3: Select a contestant to receive the THIRD prize There are 6 contestants REMAINING to choose from. So, we can complete stage 3 in 6 ways

By the Fundamental Counting Principle (FCP), we can complete all 3 stages (and thus award the 3 prizes) in (8)(7)(6) ways (= 336 ways)

Answer: 336

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(b) How many different groups of 3 people can get prizes?

In this question, the order in which we select the people doesn't matter. That is, we're just selecting 3 people. We can select 3 people from 8 people in 8C3 ways 8C3 = (8)(7)(6)/(3)(2)(1) = 56

Answer: 56

ASIDE: Here's a video on calculating combinations (like 8C3) in your head:

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