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Unformatted text preview: Review Permutations and Combinations
Some applications of counting techniques Fall 2011 EAS 305 Lecture Notes
Prof. Jun Zhuang
University at Bu↵alo, State University of New York September 9 ... 2011 Prof. Jun Zhuang Fall 2011 EAS 305 Lecture Notes Page 1 of 16 Review Permutations and Combinations
Some applications of counting techniques Agenda for Today 1 Review Permutations and Combinations 2 Some applications of counting techniques Prof. Jun Zhuang Fall 2011 EAS 305 Lecture Notes Page 2 of 16 Review Permutations and Combinations
Some applications of counting techniques Di↵erence between permutations and combinations Combinations — not concerned w/order: (a, b , c ) = (b , a, c ).
Permutations — concerned w/order: (a, b , c ) 6= (b , a, c ).
The number of permutations of n things taken r atatime is
always as least as large as the number of combinations. In
fact,. . . Prof. Jun Zhuang Fall 2011 EAS 305 Lecture Notes Page 3 of 16 Review Permutations and Combinations
Some applications of counting techniques Remark Choosing a permutation is the same as ﬁrst choosing a
combination and then putting the elements in order, i.e.,
✓◆
n!
n
=
r!
r
(n r )!
So ✓◆
n!
n
.
=
r
(n r )!r !
✓◆ ✓
◆✓◆ ✓◆
✓◆ ✓
◆
n
n
n
n
n
n
=
,
=
= 1,
=
= n.
r
nr
0
n
1
n1 Prof. Jun Zhuang Fall 2011 EAS 305 Lecture Notes Page 4 of 16 Review Permutations and Combinations
Some applications of counting techniques Examples for Combination
Example: An NBA team has 12 players. How many ways can
the coach choose the starting 5?
✓◆
12!
12
= 792.
=
5
5!7! Prof. Jun Zhuang Fall 2011 EAS 305 Lecture Notes Page 5 of 16 Review Permutations and Combinations
Some applications of counting techniques Examples for Combination
Example: An NBA team has 12 players. How many ways can
the coach choose the starting 5?
✓◆
12!
12
= 792.
=
5
5!7!
Example: Smith is one of the players on the team. How many
of the 792 starting lineups include him?
✓◆
11!
11
=
= 330.
4
4!7!
(Smith gets one of the ﬁve positions for free; there are now 4
left to be ﬁlled by the remaining 11 players.) Prof. Jun Zhuang Fall 2011 EAS 305 Lecture Notes Page 6 of 16 Review Permutations and Combinations
Some applications of counting techniques More Example Example: 7 red shoes, 5 blues. Find the number of arrangements.
R B R R B B R R R B R B I.e., how ✓ ◆ ways to put 7 reds in 12 slots?
many
12
Answer:
.
7 Prof. Jun Zhuang Fall 2011 EAS 305 Lecture Notes Page 7 of 16 Review Permutations and Combinations
Some applications of counting techniques Hypergeometric Distribution You have a objects of type 1 and b objects of type 2.
Select n objects w/o replacement from the a + b .
Pr(k type 1’s were picked)
(# ways to choose k 1’s)(choose n k 2’s)
=
# ways to choose n out of a + b
✓ ◆✓
◆
a
b
k
nk
✓
◆
=
(the hypergeometric distr’n).
a+b
n Prof. Jun Zhuang Fall 2011 EAS 305 Lecture Notes Page 8 of 16 Review Permutations and Combinations
Some applications of counting techniques Example Example: 25 sox in a box. 15 red, 10 blue. Pick 7 w/o
replacement.
✓ ◆✓ ◆
15
10
3
4
✓◆
Pr(exactly 3 reds are picked) =
25
7 Prof. Jun Zhuang Fall 2011 EAS 305 Lecture Notes Page 9 of 16 Review Permutations and Combinations
Some applications of counting techniques Permutations vs. Combinations Permutations vs. Combinations — It’s all how you approach the
problem!
Example: 4 red marbles, 2 whites. Put them in a row in random
order. Find. . .
(a) Pr(2 end marbles are W)
(b) Pr(2 end marbles aren’t both W)
(c) Pr(2 W’s are side by side) Prof. Jun Zhuang Fall 2011 EAS 305 Lecture Notes Page 10 of 16 Review Permutations and Combinations
Some applications of counting techniques Method 1 (using permutations): Let the sample space
S = {every random ordering of the 6 marbles}.
(a) A: 2 end marbles are W — WRRRRW.
A = 2!4! = 48 ) Pr(A) =
¯
(b) Pr(A) = 1 A
48
1
=
=
.
S 
720
15 Pr(A) = 14/15. Prof. Jun Zhuang Fall 2011 EAS 305 Lecture Notes Page 11 of 16 Review Permutations and Combinations
Some applications of counting techniques (c) B : 2 W’s side by side — WWRRRR or RWWRRR or . . . or
RRRRWW
B  = (# ways to select pair of slots for 2 W’s) ⇥(# ways to insert W’s into pair of slots) ⇥(# ways to insert R’s into remaining slots) = 5 ⇥ 2! ⇥ 4! = 240.
Pr(B ) = B 
240
1
=
=.
S 
720
3 But — The above method took too much time! Here’s an easier
way. . . Prof. Jun Zhuang Fall 2011 EAS 305 Lecture Notes Page 12 of 16 Review Permutations and Combinations
Some applications of counting techniques Method 2 (using combinations): Which 2 positions do the W’s occupy? Now let
S = {possible pairs of slots that the W’s occupy}.
✓◆
6
Clearly, S  =
= 15.
2
(a) Since the W’s must occupy the end slots in order for A to
occur, A = 1 ) Pr(A) = A/S  = 1/15.
¯
(b) Pr(A) = 14/15.
(c) B  = 5 ) Pr(B ) = 5/15 = 1/3. Prof. Jun Zhuang Fall 2011 EAS 305 Lecture Notes Page 13 of 16 Review Permutations and Combinations
Some applications of counting techniques Birthday Problem
n people in a room. Find the prob that at least two have the same
birthday. (Ignore Feb. 29, and assume that all 365 days have equal
prob.)
A: All birthdays are di↵erent.
S = {(x1 , . . . , xn ) : xi = 1, 2, . . . , 365} (xi is person i ’s birthday),
and note that S  = (365)n .
A = P365,n = (365)(364) · · · (365 n + 1) (365)(364) · · · (365 n + 1)
(365)n
364 363
365 n + 1
= 1·
·
···
365 365
365 Pr(A) = Prof. Jun Zhuang Fall 2011 EAS 305 Lecture Notes Page 14 of 16 Review Permutations and Combinations
Some applications of counting techniques We want
¯
Pr(A) = 1 ✓ 364 363
365 n + 1
1·
·
···
365 365
365 ¯
Notes: When n = 366, Pr(A) = 1.
¯
For Pr(A) to be > 1/2, n must be
¯
When n = 50, Pr(A) = 0.97. Prof. Jun Zhuang ◆ 23. (surprising) Fall 2011 EAS 305 Lecture Notes Page 15 of 16 Review Permutations and Combinations
Some applications of counting techniques Multinomial Coe cients Example: n1 blue sox, n2 reds. # of assortments is ✓ n1 + n2
n1 (binomial coe cients).
P
Generalization (for k types of objects): n = k=1 ni
i
# of arrangements is n1 !n2n!!···nk ! .
Example: How many ways can “Mississippi” be arranged? ◆ # perm’s of 11 letters
11!
=
= 34, 650.
(# m’s)!(# p ’s)!(# i ’s)!(# s ’s)!
1!2!4!4! Prof. Jun Zhuang Fall 2011 EAS 305 Lecture Notes Page 16 of 16 ...
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This note was uploaded on 09/25/2011 for the course ECON 101 taught by Professor Wang during the Spring '11 term at SUNY Buffalo.
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