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Chapter 1: Problem 14
How many 5 -card poker hands are there?
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A president, treasurer, and secretary, all different, are to be chosen from a club consisting of 10 people. How many different choices of officers are possible if (a) there are no restrictions; (b) \(A\) and \(B\) will not serve together; (c) \(C\) and \(D\) will serve together or not at all; (d) \(E\) must be an officer; (e) \(F\) will serve only if he is president?
Consider a function \(f\left(x_{1}, \ldots, x_{n}\right)\) of \(n\) variables. How many different partial derivatives of order \(r\) does it possess?
A committee of 7, consisting of 2 Republicans, 2 Democrats, and 3 Independents, is to be chosen from a group of 5 Republicans, 6 Democrats, and 4 Independents. How many committees are possible?
Consider three classes, each consisting of \(n\) students. From this group of \(3 n\) students, a group of 3 students is to be chosen. (a) How many choices are possible? (b) How many choices are there in which all 3 students are in the same class? (c) How many choices are there in which 2 of the 3 students are in the same class and the other student is in a different class? (d) How many choices are there in which all 3 students are in different classes? (e) Using the results of parts (a) through (d), write a combinatorial identity.
How many outcome sequences are possible when a die is rolled four times, where we say, for instance, that the outcome is \(3,4,3,1\) if the first roll landed on 3 , the second on 4 , the third on 3 , and the fourth on \(1 ?\)
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