CMSC250
Combinatorics
Combinatorics
Counting
Combinatorics: hard to define, but has to do with counting
Counting is easy
Counting is easy
How many letters in the English Alphabet?
How many characters in the string "COUNT"?
How many 3 character strings can be made from "COUNT"?
Counting is (somewhat) easy
How many letters in the English Alphabet?
How many characters in the string "COUNT"?
How many 3 character strings can be made from "COUNT"?
Counting is (somewhat) easy
How many values can 3 bits represent?
How many values can a byte represent?
How many values can a byte with 3 '1's represent?
Counting is (somewhat) easy hard
How many values can 3 bits represent?
How many values can a byte represent?
How many values can a byte with 3 '1's represent?
Counting
Counting is (somewhat) easy hard
But we have rules to help make it easy
Note: we will be counting events, or actions
Multiplication rule of Counting
If we have some event $A$ which has $x$ number of ways of occurring, and we have some event $B$ which has $y$ number of ways of occurring, then there are $xy$ ways of both A and B occurring
Multiplication rule of Counting
Event 1: Flip a coin, Event 2: roll a 6-sided dice
How many outcomes are possible?
$2 \cdot 6$
$\{H1,H2,H3,H4,H5,H6,T1,T2,T3,T4,T5,T6\}$
Multiplication rule of Counting
Event 1: Choose a letter of the alphabet
Event 2: Choose a letter of the alphabet
How many 2 character strings exist?
$26 \cdot 26$
Multiplication rule of Counting
Event 1: Choose a letter of the alphabet
Event 2: Choose a letter of the alphabet
Event 3: Choose a letter of the alphabet
How many 3 character strings exist?
$26 \cdot 26 \cdot 26$
Multiplication rule of Counting
How many 4-bit strings exist?
$2 \cdot 2 \cdot 2 \cdot 2$
Addition rule of Counting
If we have some event $A$ which has $x$ number of ways of occurring, and we have some event $B$ which has $y$ number of ways of occurring, and only one event can occur, then there are $x + y$ ways of either (but not both) events occurring
Addition rule of Counting
Event 1: Flip a coin, Event 2: roll a 6-sided dice
How many outcomes are possible?
$2 + 6$
$\{H,T,1,2,3,4,5,6\}$
Addition rule of Counting
Event 1: Choose a letter, Event 2: Choose a digit
How many outcomes are possible?
$26 + 10$
Event 1: Choose an alphanumeric value
Event 2: Choose an alphanumeric value
How many Alphanumeric Strings of size 2?
$(26 + 10) \cdot (26 + 10)$
How many alphabetic strings of length 2 and 4 exist?
$(26 \cdot 26) + (26 \cdot 26 \cdot 26 \cdot 26)$
Permutations
Recall: How many characters in the string "COUNT"?
How many strings are anagrams of "COUNT"?
_ _ _ _ _
5 options for first character
Recall: How many characters in the string "COUNT"?
How many strings are anagrams of "COUNT"?
T _ _ _ _
4 options for second character
Recall: How many characters in the string "COUNT"?
How many strings are anagrams of "COUNT"?
T O _ _ _
3 options for third character
Recall: How many characters in the string "COUNT"?
How many strings are anagrams of "COUNT"?
T O N _ _
2 options for fourth character
Recall: How many characters in the string "COUNT"?
How many strings are anagrams of "COUNT"?
T O N C _
1 options for fifth character
Recall: How many characters in the string "COUNT"?
How many strings are anagrams of "COUNT"?
T O N C U
Recall: How many characters in the string "COUNT"?
How many strings are anagrams of "COUNT"?
T O N C U
$5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 5!$
Recall: How many characters in the string "COUNT"?
How many strings are anagrams of "COUNT"?
T O N C U
$5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 5!$
Assumptions: no repetitions, use all characters
How many 3 character strings can be made from "COUNT"?
_ _ _
5 options for first character
How many 3 character strings can be made from "COUNT"?
T _ _
4 options for second character
How many 3 character strings can be made from "COUNT"?
T O _
3 options for third character
How many 3 character strings can be made from "COUNT"?
T O N
How many 3 character strings can be made from "COUNT"?
T O N
$5 \cdot 4 \cdot 3$
How many 4 character strings can be made from "COUNT"?
_ _ _ _
$5 \cdot 4 \cdot 3 \cdot 2$
How many 2 character strings can be made from "COUNT"?
_ _
$5 \cdot 4$
How many 2 character strings can be made from "COUNT"?
_ _
$5 \cdot 4$
Pattern?
Pattern?
4 Character strings = $5 \cdot 4 \cdot 3 \cdot 2$
3 Character strings = $5 \cdot 4 \cdot 3$
2 Character strings = $5 \cdot 4$
Arrangements of size $k$ taken from a set of size $n$ can be represented as
$\frac{n!}{(n-k)!} = P(n,k)$
Arrangements of size $k$ taken from a set of size $n$ can be represented as
$\frac{n!}{(n-k!)} = P(n,k)$
Note: No repetitions: "CCO" is not allowed, and all characters in "COUNT" are unique
Note: Order matters: "COU" $\neq$ "OUC"
Examples
How many ATM Pins exist?: $10^4$
How many ATM Pins with no repeated digits exist?: $\frac{10!}{(10-4)!}$
How many 3 character strings of "Justin" exist?: $\frac{6!}{(6-3)!}$
How many 5 unique character strings exist?$\frac{26!}{(26-5)!}$
Combinations
What if order did not matter?
How many values can a byte with 3 '1's represent?
How many anagrams from the string: $00000111$
However: $1_01_11_20_00_10_20_30_4 = 1_11_01_20_00_10_20_30_4$
What if order did not matter?
How many values can a byte with 3 '1's represent?
How many anagrams from the string: $00000111$
$00000111 = ABCDEFGH$
However: $1_01_11_20_00_10_20_30_4 = 1_11_01_20_00_10_20_30_4$
However: $FGHABCDE = GFHABCDE$
How many values can a byte with 3 '1's represent?
$00000111 = ABCDEFGH$
However: $FGHABCDE = GFHABCDE$
$FGHx,FHGx,GFHx,GHFx,HFGx,HGFx$, where $x= ABCDE$, is the same
Permutations of $FGH = 3!$
How many values can a byte with 3 '1's represent?
$00000111 = ABCDEFGH$
Permutations of $FGH = 3!$
The Number of subsets of size $k$ from a set of size $n$ can be represented as
$\frac{n!}{k!(n-k)!} = \frac{P(n,k)}{k!} = C(n,k) = {n \choose k}$
Examples
How many ways can I create a group of 4 from 10 people?: $C(10,4)$
How many bytes with 6 '1's exist?: $C(8,6)$
${n \choose 1}$: n
${n \choose n}$: 1
12 people: forming a team of 5, but 2 people stick together?: $C(10,3) + C(10,5)$