ICS 6D Sandy Irani Two Different Counting Problems Student council has 15 members Must select officers Pres VP Treasurer Secretary Sample selection Sally Frank Margaret George ID: 561942
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Slide1
Counting Subsets
ICS 6D
Sandy
IraniSlide2
Two Different Counting Problems
Student council has 15 members. Must select officers (
Pres
, VP, Treasurer, Secretary
)
Sample selection: (Sally, Frank, Margaret, George)
A selection is a 4-permutation
Student council has 15 members.
Must select an executive committee with 4 members.
Sample selection:
{Sally
, Frank, Margaret,
George}
= {
Margaret,
Sally
, Frank,
George}
A selection is a
4-subsetSlide3
Counting Subsets vs. Permutations
S = {a, b, c}
The number of 2-permutations from S is 6:
(a, b), (a, c), (b, a), (b, c), (c, a), (c, b)
The number of 2-subsets from S is 3:
{a, b}, {a, c}, {b, c}Slide4
Counting Subsets
S = {blue, green, orange, pink, red}
How many ways to pick a subset of 3 colors?
Know how to count 3-permutations.
f:
3-permutations from S
→
3-subsets from S
f(
(blue, pink, green)
) =
{blue, green, pink}
f(
(pink
,
blue, green
)
) =
{blue, green, pink}Slide5
Counting Subsets
f:
3-permutations from S
→
3-subsets from S
Function f is 3!-to-1Slide6
Counting Subsets
Set S with n elements
f:
r-permutations from S
→
r-subsets from S
Function f is r!-to-1Slide7
Counting Subsets
The number of r-subsets chosen from a set of n elements is
=
“n choose r”
Slide8
Calculating n choose r
Slide9
Fact:
Can check by arithmetic:Slide10
Fact:
Can show by bijection. S = {1, 2, 3,…, n}
f: r-subsets of S
(n-r)-subsets of S
f(A) = A
Example: S = {1, 2, 3, 4, 5, 6, 7, 8} n = 8, r = 3
f is a bijection: f
-1
(B) = B
Slide11
Subsets vs. Permutations
100 pianists compete in a piano competition.
In the first round 25 of the 100 contestants are selected to go on to the next round. How many different possible outcomes are there?
In the second round, the judges select a first, second, third, fourth and fifth place winners
of the
competition from among
the 25
pianists
who advanced to the second round. How
many outcomes
are there for the second round of the
competition?Slide12
Counting Strings
How many binary strings with 9 bits have exactly four 1’s?
By
bijection
: S = {1, 2, 3, 4, 5, 6, 7, 8, 9}
g: 4-subsets of S
9-bit strings with four 1’s
g(A) = y if
jA
, then
j
th
bit of y is 1
if jA, then jth bit of y is 0g( {1, 3, 4, 9} ) =g
-1
(010011010) = Slide13
Counting Strings
Strings over the alphabet {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
How many strings of length 8?
How many strings of length 8 have exactly three 5’s? Slide14
Counting Strings
Strings over the alphabet {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
How many strings of length 8 have exactly three 5’s and two 1’s? Slide15
Distribution Problems
How many ways are there to distribute 10
identical
prizes to 275 people with at most one per person?
The students are all different (distinguishable)
“indistinguishable” = “identical”
How many ways are there to distribute 10
different
prizes to 275 people with at most one per person?Slide16
Distribution Problems
How many ways are there to distribute 10
different
prizes to 275 people with no limit on the number of prizes per person?Slide17
Playing Cards
Standard deck of playing cards has 52 cards
13 ranks: A
, 2
, 3, 4, 5, 6, 7, 8, 9, J, Q, K
4 suits:
Each card has a rank and a suit:
8
Q
Every rank/suit combination possible: # cards = # ranks · # suits = 4 · 13 = 52Slide18
5-card Hand
A 5-card hand is a subset of 5 of the 52 cards:
{
A
,
A, 4,
2
,
8
}
(order doesn’t matter)
How many different 5-card hands are there from a standard playing deck? Slide19
5-card Hands
How many 5-card hands have exactly 3 clubs?Slide20
5-card Hands
How many 5-card hands are a 3-of-a-kind?
(3 cards of the same rank, the two other cards have a different rank from the 3-of-a-kind and from each other)Slide21
5-card Hands
How many 5-card hands have two pairs?
(Each pair has the same rank. Two pairs have different rank from each other. The 5
th
card has a different rank than the two pairs)Slide22
5-card Hands
How many 5-card hands have no face cards?
(No A, J, Q, K)Slide23
More Counting
A club has 10 men and 9 women
How many ways are there to select a committee of 6 people from the club members?
Same number of men as women