Questions that ask, “How many ways can people be lined up for a photograph” or “how many five-card hands have two aces” or “how many ways can you fill out a multiple choice test” are all counting problems. <\/p>\n
When an item, once used, cannot be used again, we call that without replacement.<\/em> Counting without replacement can often be done using permute or choose, but there are other options. \u00a0For a quick summary of permute and choose, see my article here.<\/a><\/p>\n When an item can be used over again, we call that with replacement.<\/em> A question with replacement is often done using an exponential expression, as we shall see.<\/p>\n Example 1: How many ways can you arrange the letters of the word radio?<\/strong><\/p>\n Solution: The answer is 5P5, or 120. You use P – which means Permute – because the order of the letters is what matters, and because the process is without replacement: that is, in a given arrangement, each letter can only appear once.<\/p>\n Example 2: How many five card hands from a standard deck of 52 cards have exactly 2 aces?<\/strong><\/p>\n Solution: Here, the order doesn’t matter, but the process is still without replacement: if you get the queen of hearts, you cannot draw it again in the same hand. Without replacement, and where order doesn’t matter, points to Choose (aka Combination): Specifically, (4C2)(48C3) or 103776. The reasoning is as follows: from the 4 aces, choose 2: 4C2. Then, from the other 48 non-ace cards, choose 3: 48C3. Multiply the two results together since they both happen yet are independent of each other. Independent means that the choosing of the two aces has no influence on which non-ace cards will be chosen.<\/p>\n Example 3: How many possible ways can a 20-question multiple choice test be filled out, if each question has five options? I’ll say more about counting in future posts. Hope your exams are going well:)<\/p>\n
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\nSolution: Order matters here, but it is with replacement<\/em>: for instance, you can choose answer A over and over again. The number of ways you can fill out the test is 5^20. The reasoning is that you can fill out the first question five possible ways, then the second five possible ways, and so on. Your possibilities are (5)(5)(5)(5)……(5)=5^20. The number is absurdly large, so we’ll just refer to it as 5^20.<\/p>\n