Chapter 2, Problem 43E

Consider a weighted voting system with six players ( $P_1$ through $P_6$).

a. Find the total number of coalitions in this weighted voting system.

b. How many coalitions in this weighted voting system do not include $P_1$ (Hint: Think of all the possible coalitions of the remaining players.)

c. How many coalitions in this weighted voting system do not include $P_3$? [Hint: Is this really different from (b)?]

d. How many coalitions in this weighted voting system do not include $P_1\text{ and }{P}_3$?

e. How many coalitions in this weighted voting system include both $P_1\text{ and }{P}_3$? [Hint: Use your answers for (a) and (d).]