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