PRINCIPLE OF CHOICE
The principle of choice is discussed in relation to combinations, although it is also discussed later in this chapter in relation to permutations. It is stated as follows:
If a selection can be made in n, ways, and after this selection is made, a second selection can be made in nz ways; and after this selection is made, a third selection can be made in n, ways; and so forth for r selections, then the r selections can be made together in
EXAMPLE: In how many ways can a coach choose first a football team and then a basketball team from 18 boys?
SOLUTION: First let the coach choose a football team; that is,
The coach now must choose a basketball team from the remaining seven boys; that is,
Then, together, the two teams can be chosen in
(31,824)(21) = 668,304 ways
The same answer would be achieved if the coach chose the basket
ball team first and then the football team; that is,
which is the same number as before.
EXAMPLE: A woman ordering dinner has a choice of one meat dish from four, four vegetables from seven, one salad from three, and one dessert from four. How many different menus are possible?
SOLUTION: The individual combinations are as follows:
The values of these combinations are
Therefore, the woman has a choice of
(4)(35)(3)(4) = 1,680
Solve the following problems:
1. A man has 12 different colored shirts and 20 different ties. How many shirt and tie combinations can he select to take on a trip if he takes 3 shirts and 5 ties?
2. A petty officer in charge of posting the watch has 12 men in his duty section. He must post 3 different fire watches and then post 4 aircraft guards on different aircraft. How many different assignments of men can he make?
3. If 10 third class and 14 second class petty officers are in a division that must furnish 2 second class and 6 third class petty officers for shore patrol, how many different shore patrol parties can be made?