91Ó°ÊÓ

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?

From a group of 8 women and 6 men a committee consisting of 3 men and 3 women is to be formed. How many different committees are possible if (a) 2 of the men refuse to serve together; (b) 2 of the women refuse to serve together; (c) 1 man and 1 woman refus? to serve together?

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?

The game of bridge is played by 4 players, each of whom is dealt 13 cards. How many bridge deals are possible?

If 12 people are to be divided into 3 committees of respective sizes 3,4, and 5 , how many divisions are possible?

Delegates from 10 countries, including Russia, France, England, and the United States, are to be seated in a row. How many different seating arrangements are possible if the French and English delegates are to be seated next to each other, and the Russian and U.S. delegates are not to be next to each other?

An elevator starts at the basement with 8 people (not including the elevator operator) and discharges them all by the time it reaches the top floor, number 6. In how many ways could the operator have perceived the people leaving the elevator if all people look alike to him? What if the 8 people consisted of 5 men and 3 women and the operator could tell a man from a woman?

We have 20 thousand dollars that must be invested among 4 possible opportunities. Each investment must be integral in units of 1 thousand dollars, and there are minimal investments that need to be made if one is to invest in these opportunities. The minimal investments are \(2,2,3\), and 4 thousand dollars. How many different investment strategies are available if (a) an investment must be made in each opportunity; (b) investments must be made in at least 3 of the 4 opportunities?