GMAT Official Guide Quantitative Review 2019_ Book

GMAT® Official Guide 2019 Quantitative Review

Thus, by the multiplication principle, the number of ways of ordering the n objects is

n(n -l)(n-2) · · ·(3)(2)(1) = n!.

For example, the number of ways of ordering the letters A, B, and C is 3!, or 6:

ABC, ACB, BAC, BCA, CAB, and CBA.

These orderings are called the permutations of the letters A, B, and C.

A permutation can be thought of as a selection process in which objects are selected one by one in a

certain order. If the order of selection is not relevant and only k objects are to be selected from a larger

set of n objects, a different counting method is employed.

Specifically, consider a set of n objects from which a complete selection of k objects is to be made

without regard to order, where O S k S n. Then the number of possible complete selections of k objects is

called the number of combinations of n objects taken k at a time and is denoted by ( ~).

The value of (nk) is given by (nk) = n!.

k!(n-k)!

Note that ( ~) is the number of k-element subsets of a set with n elements. For example,

if S = [A, B, C, D, E}, then the number of 2-element subsets of S, or the number of combinations of

(^51) etters ta k en (^2) at a time,.. 1s (5)^5!^120 10
2 = 2! 3! = ( 2 ) ( 6 ) =.
The subsets are [A, B}, [A, C}, [A, D}, [A, E}, [B, C}, [B, D}, [B, E}, [C, D}, [C, E}, and [D, E}. Note
that ( ~) = 10 = ( ~) because every 2-element subset chosen from a set of 5 elements corresponds to a
unique 3-element subset consisting of the elements not chosen.

11. Discrete Probability
    Many of the ideas discussed in the preceding three topics are important to the study of discrete
    probability. Discrete probability is concerned with experiments that have a finite number of outcomes.
    Given such an experiment, an event is a particular set of outcomes. For example, rolling a number cube
    with faces numbered 1 to 6 (similar to a 6-sided die) is an experiment with 6 possible outcomes:
    1, 2, 3, 4, 5, or 6. One event in this experiment is that the outcome is 4, denoted [ 4}; another event is
    that the outcome is an odd number: [1, 3, 5}.

The probability that an event E occurs, denoted by P(E), is a number between O and 1, inclusive.

If E has no outcomes, then Eis impossible and P(E) = O; if Eis the set of all possible outcomes of the

experiment, then Eis certain to occur and P(E) = 1. Otherwise, Eis possible but uncertain, and

0 < P(E) < 1. If Fis a subset of E, then P(F) S P(E). In the example above, if the probability of each of

the 6 outcomes is the same, then the probability of each outcome is 1.. , and the outcomes are said to be

6