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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

different menus.


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?


1. 3,410,880

2. 27,720

3. 19,110

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