A student club consists of 3 sophomores, 5 juniors, and 6 seniors. A committee o

Question

A student club consists of 3 sophomores, 5 juniors, and 6 seniors. A committee of size 6 will be formed (by random selection). What is the probability that the committee contains seniors only? What is the probability that the committee contains all 3 sophomores and at least one junior and at least one senior? What is the probability that the committee contains no seniors? What is the probability that the committee contains exactly 4 juniors? One of the sophomores is the friend of one of the juniors. What is the probability that both arc on the committee? What is the probability that neither arc on the committee? What is the probability that only one (or the other) of them is on the committee?

Explanation / Answer

Whenever you're dealing with a problem of forming sub-groupsout of a larger pool, you should think of usingcombinatorics. I assume, if you're getting questions likethis, that your class has covered the formula for counting thenumber of combinations; the trick is figuring out how to deployit. . By random selection, I further assume that they mean everypossible combination has an equal probability of being selected(though they should really have spelled that out). . So first you need to figure out the total number of possiblecombinations of students they could put together. There are14 students, and they're picking 6 of 14 (withoutreplacement). If there are n combinations you can get, theneach combination has a 1/n chance of being drawn. . Once you've done that, you need to find, for each part, howmany combinations meet the various conditions. There isexactly one combination that calls for using all 6 seniors. Other conditions are harder, and often require you to deal withdifferent conditions separately and then multiply them together (oradd, as required by logic). For example, in part b, you startby taking all 3 sophomores. Boom, done. That leaves 3other slots, and you need to find all the combinations that involvepicking exactly 1 junior OR exactly 2 juniors. (Those will bethe combinations that have at least 1 junior and 1 senior onit.)  So there's a case where you'll consider differentcases and add them up.

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