Counting_and_Probability_Distributions_stud.doc
Feb 24’10
PERMUTATIONS AND COMBINATIONS
(handout #6;
03/15/09)
THE COUNTING (MULTIPLICATION) PRINCIPLE
.
If one activity can occur in
any of
m
ways, and,
following this
, a second activity can occur in any of
n
ways, then
both activities can occur in the order given in
m*n
ways.
Example
. Let us consider a bank containing 4 coins: a quarter (
Q
), a dime (
D
), a nickel (
N
), and
a penny (
P
). If one coin is taken out of the bank at random (the first activity), then the set of
4
possible outcomes (sample space) is
{Q, D, N, P }
. If a coin is tossed (the second activity), there
are
2
possible outcomes, heads (
h
) and tails (
t
), and the sample space is
{h, t} .
If a coin is taken
out of the bank and tossed, then the sample space is the set of
4*2 = 8
possible outcomes:
{ (Q, h), (Q, t);
(D, h), (D, t);
(N, h), (N, t);
(P, h), (P, t) }
Problem 1
. A bank contains
4
different coins: a quarter (
Q
), a dime (
D
), a nickel (
N
),
and a penny (
P
), which are to be drawn out one at a time
without replacement
. In how
many
different orders
can the
4
coins be removed from the bank?
Solution
. There are 4 possible outcomes for the first draw:
Q, D, N, P.

Then, since 3 coins remain in the bank,
 Number of Coins  Number of

there are 3 possible coins to be drawn on
Draw  in the Bank
 Possible Outcomes
the 2
nd
draw, and in accordance with the

counting principle there are
4*3 = 12
1
st

4

4

possible orders in which the first two
2
nd

3

4*3

coins
could be removed:
3
rd

2

4*3*2

(Q,
D),
(Q, N),
(Q, P);
(D,
Q),
(D, N), (D, P);
4
th

1

4*3*2*1

(N, Q),
(N, D),
(N, P);
(P,
Q)
,
(P,
D),
(P, N).
On the 3
rd
draw there are 2 coins left to be selected and
4*3*2 = 24
possible orders in
which the first three
coins could be removed:
(Q,
D, N),
(Q,
D, P);
(Q, N, D),
(Q, N, P);
(Q, P, D),
(Q, P, N);
(D,
Q, N),
(D, Q, P);
(D, N, Q),
(D, N, P);
(D, P, Q),
(D, P, N);
(N, Q, D),
(N, Q, P);
(N, D, Q),
(N, D, P);
(N, P, Q),
(N, P, D);
(P,
Q, D)
,
(P, Q, N);
(P, D, Q),
(P, D, N);
(P, N, Q),
(P, N, D).
On the last draw there is one coin left to be selected and
4*3*2*1 = 24
possible orders in
which the coins could have been drawn.
GENERAL COUNTING (MULTIPLICATION) PRINCIPLE
.
If an activity
1
can
occur in any of
n
1
ways, and,
following this
, an activity
2
can occur in any of
n
2
ways,
an activity
3
can occur in any of
n
3
ways, and so on,
then all these activities can occur
in
n
1
*
n
2
*
n
3
…
different ways.
1
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DEFINITION
.
A permutation
is an arrangement of objects in some
specific order
. In
general, the number of permutations of
n
things (elements) taken
r
at a time
n
P
r
= n
*
(n – 1)
*
(n – 2)
*
…* (n – r + 1) = n!/(n – r)!
,
(r ≤ n)
(6.1)
\_________
r
factors
_________/
In (6.1)
n!
= 1
*
2
*
3
*
…
*
n
means
the factorial
of
n
; by definition of the factorial
,
0! = 1
.
For numerical evaluations of the number of permutations
n
P
r
the following representation
of
n!
may be useful:
n!
= 1
*
2
*
3
*
…
*
n = (n – 1)!
*
n = (n – 2)!
*
(n – 1)*n = (n – 3)!
*
(n – 2)
*
(n – 1)*n
(6.1a)
and so on.
(
CAUTION
. The letter
P
represents here
permutations
, not
probability
.)
In the special case, when
r = n
,
the number of permutations of
n
things taken
n
at a
time
n
P
n
= n
*
(n – 1)
*
(n – 2)
*
…
*
1 = n!
(6.2)
In terms of permutations, the number of orders in which
2
coins can be drawn out of
the bank containing
4
different
coins in our problem 1 is
4
P
2
= 4*3 = 12
[see (6.1)]. {The
symbol
4
P
2
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 Spring '05
 Kzaer
 Permutations, Multiplication, Counting, Permutations And Combinations, Probability, Probability distribution, Probability theory, NPR

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