<p>Leticia plans to display four of nine stamps in a special frame. In how many orders can she display the stamps?</p>
<p>The answer is 9!/5! but I need a clear explanation. Thank you!</p>
<p>Also:
A fair coin is flipped five times in succession. What is the probability that there are at least one head and one tail?</p>
<p>@letjustin</p>
<p>If you can do the following problem then I would recommend you to see the link between this and the nine stamps problem:</p>
<p>How many distinct 7 letter words(may not have any meaning) can be formed by rearranging the letters in the word PENNANT? </p>
<p>The key is to understand how to deal with the letter N.</p>
<p>@letjustin</p>
<p>For the coin problem think of the event that is complementary to at least one head and one tail outcome.</p>
<p>^ SATQuantum’s hint is on the money. It’s actually TWO events that you need to think of, but they are equally likely. And for the record, this is harder than an SAT problem, but it is interesting. If you don’t think of the way SATQuantum is hinting at, you can also work out the right answer by working through separate cases: getting exactly one head, exactly two heads, exactly three and then exactly 4…not pretty, but not impossible.</p>
<p>Yeah, in some problems, complementary counting solves much faster than other methods. If the problem seems like it’ll take a long time by direct counting, see if complementary counting can lead to a faster solution and if so, do it.</p>
<p>For SATQuantum’s first problem, I think the answer is 7!/3! to deal with the N because some words will be repeated. For the coin problem, I think it can be solved by subtracting from 1 the probability that there is not at least 1 tail or 1 head, which means they are all heads or all tails. There are only 2 cases of that, so you subtract 2 from 32 (2^5) and you get 15/16. Thanks for your time everyone!</p>
<p>The first one:</p>
<p>It is a permutation, because the stamps don’t have to be in a specific order (if you are displaying stamps 1,2,3,4… they could be in the order 1,2,3,4, or 1,3,2,4, or ,1,4,3,2, etc…) So just plug into your calculator 9 Permutation 4, and you get your answer. Or, if you don’t have a scientific calculator (or any calculator for that matter) you could just memorize the formulas for permutations and combinations.</p>
<p>The second one:</p>
<p>There are only two ways that don’t satisfy the need to have one head and one tails (either all heads, or all tails), so then you calculate the amount of other orders. For the first coin, it could be either two choices (heads or tails), then for the second one you have two choices (heads or tails), and so on for all 5 flips, so you just multiply 2x2x2x2x2 which is 32, then you just divide 30 by 32 to get the probability, which is 15/16, which is 93.75%</p>
<p>If you ever need help with other questions, you could DM me (I got 1 wrong both times I took the SAT (and the first time, it said I ommitted one (which I didn’t), so technically, I should have gotten an 800)</p>