Probability And Statistics Worksheets Page 25
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42For each situation described in the preceding Examples, the order
Math: Who,
of the items in consideration is important. A set of items in no
Where, When?
particular order is called a combination. The number of
combinations of n different items, taken r items at a time, where
Although several 16th-
0
r
n, is written
C
. You can use the formula below to find the
n
r
and 17th-century
number of combinations of a set of items.
mathematicians, notably
n!
Pascal and Fermat,
C
n
r
investigated probability,
(n
r)!r!
Jacques Bernoulli is
considered by some to be
E x a m p l e 4
the founder of probability
theory. His book, Ars
Conjectandi, published in
How many different four-person ensembles can be chosen from a
1713, is the first book
pool of 10 musicians?
devoted entirely to the
subject of probability. This
book contains a theory of
Solution
combinations, essentially
the same as we
There are 10 people from which to pick, 4 at a time. So, n
10 and
understand it today, as
r
4. Use the formula:
well as the first
appearance, with today’s
n!
C
meaning, of the word
n
r
(n
r)!r!
permutation.
10!
C
10
4
(10
4)!4!
10 9 8 7 6 5 4 3 2 1
(6 5 4 3 2 1)(4 3 2 1)
Cancel common factors to simplify the
5
0
40
computation.
2
4
210
There can be 210 different four-person ensembles.
T
T
E
RY
HESE
XERCISES
1.
SPORTS
A manager is choosing her infield from among 4 third-base players,
3 shortstops, 2 second-base players, and 5 first-base players. How many
different ways can an infield be chosen?
Tell whether each question involves a permutation or a combination. Then solve.
2. In how many different ways can you arrange the letters a, c, e, g, i, k, and 1?
3. How many different selections of three tapes can be made by a consumer
choosing from among a collection of six tapes?
4. In how many different ways can a starting lineup of 5 players be selected
from a group of 12 basketball players?
5. In how many different ways can 4 winners be chosen from 15 contestants?
P
E
RACTICE
XERCISES • For Extra Practice, see page 691.
6.
OFFICE WORK
A secretary has to create ten new customer files. In how
many different orders can he do this?
7.
HIRING
Six applicants apply for two jobs. How many different outcomes are
possible?
404
Chapter 9 Probability and Statistics
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