Permutations. CS311H: Discrete Mathematics. Permutations and Combinations. How Many Permutations? Examples. Computing P(n, r) r-permutations
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1 Pemutations CS311H: Discete Mathematics Pemutations and Combinations Instucto: Işıl Dillig A pemutation of a set of distinct objects is an odeed aangement of these objects No object can be selected moe than once Ode of aangement mattes Example: S {a, b, c}. What ae the pemutations of S? Instucto: Işıl Dillig, CS311H: Discete Mathematics Pemutations and Combinations 1/36 Instucto: Işıl Dillig, CS311H: Discete Mathematics Pemutations and Combinations 2/36 How Many Pemutations? Examples Conside set S {a 1, a 2,... a n } How many pemutations of S ae thee? Decompose using poduct ule: How many ways to choose fist element? How many ways to choose second element?... Conside the set {7, 10, 23, 4}. How many pemutations? How many pemutations of lettes A, B, C, D, E, F, G contain ABC as a substing? How many ways to choose last element? What is numbe of pemutations of set S? Instucto: Işıl Dillig, CS311H: Discete Mathematics Pemutations and Combinations 3/36 Instucto: Işıl Dillig, CS311H: Discete Mathematics Pemutations and Combinations 4/36 -Pemutations Computing P(n, -pemutation is odeed aangment of elements in a set S S can contain moe than elements But we want aangement containing of the elements in S The numbe of -pemutations in a set with n elements is witten P(n, Example: What is P(n, n? Given a set with n elements, what is P(n,? Decompose using poduct ule: How many ways to pic fist element? How many ways to pic second element? How many ways to pic i th element? How many ways to pic last element? Thus, P(n, n (n 1... (n 1 n! (n! Instucto: Işıl Dillig, CS311H: Discete Mathematics Pemutations and Combinations 5/36 Instucto: Işıl Dillig, CS311H: Discete Mathematics Pemutations and Combinations 6/36 1
2 Examples Combinations What is the numbe of 2-pemutations of set {a, b, c, d, e}? How many ways to select fist-pize winne, second-pize winne, thid-pize winne fom 10 people in a contest? Salesman must visit 4 cities fom list of 10 cities: Must begin in Chicago, but can choose the emaining cities and ode. An -combination of set S is the unodeed selection of elements fom that set Unlie pemutations, ode does not matte in combinations Example: What ae 2-combinations of the set {a, b, c}? Fo this set, 6 2-pemutations, but only 3 2-combinations How many possible itineay choices ae thee? Instucto: Işıl Dillig, CS311H: Discete Mathematics Pemutations and Combinations 7/36 Instucto: Işıl Dillig, CS311H: Discete Mathematics Pemutations and Combinations 8/36 Numbe of -combinations The numbe of -combinations of a set with n elements is witten C (n, C (n, is often also witten as Theoem: is also called the binomial coefficient, ead n choose ( n n! C (n,! (n! Poof of Theoem What is the elationship between P(n, and C (n,? Let s decompose P(n, using poduct ule: Fist choose elements Then, ode these elements How many ways to choose elements fom n? How many ways to ode elements? Thus, P(n, C (n,! Theefoe, C (n, P(n,! n! (n!! Instucto: Işıl Dillig, CS311H: Discete Mathematics Pemutations and Combinations 9/36 Instucto: Işıl Dillig, CS311H: Discete Mathematics Pemutations and Combinations 10/36 Examples Moe Complicated Example How many hands of 5 cads can be dealt fom a standad dec of 52 cads? Thee ae 9 faculty membes in a math depatment, and 11 in CS depatment. If we must select 3 math and 4 CS faculty fo a committee, how many ways ae thee to fom this committee? How many bitstings of length 8 contain at least 6 ones? Instucto: Işıl Dillig, CS311H: Discete Mathematics Pemutations and Combinations 11/36 Instucto: Işıl Dillig, CS311H: Discete Mathematics Pemutations and Combinations 12/36 2
3 One Moe Example Binomial Coefficients How many bitstings of length 8 contain at least 3 ones and 3 zeos? Recall: C (n, is also denoted as binomial coefficient and is called the Binomial is polynomial with two tems, e.g., (a b, (a b 2 ( n called binomial coefficient b/c it occus as coefficients in the expansion of (a b n Instucto: Işıl Dillig, CS311H: Discete Mathematics Pemutations and Combinations 13/36 Instucto: Işıl Dillig, CS311H: Discete Mathematics Pemutations and Combinations 14/36 An Example The Binomial Theoem Conside expansion of (a b 3 (a b 3 (a b(a b(a b (a 2 2ab b 2 (a b (a 3 2a 2 b ab 2 (a 2 b 2ab 2 b 3 1a 3 3a 2 b 3ab 2 1b 3 Let x, y be vaiables and n a non-negative intege. Then, (x y n n j 0 j What is the expansion of (x y 4? x n j y j Instucto: Işıl Dillig, CS311H: Discete Mathematics Pemutations and Combinations 15/36 Instucto: Işıl Dillig, CS311H: Discete Mathematics Pemutations and Combinations 16/36 Anothe Example Coollay of Binomial Theoem What is the coefficient of x 12 y 13 in the expansion of (2x 3y 25? Binomial theoem allows showing a bunch of useful esults. Coollay: n 0 2 n Instucto: Işıl Dillig, CS311H: Discete Mathematics Pemutations and Combinations 17/36 Instucto: Işıl Dillig, CS311H: Discete Mathematics Pemutations and Combinations 18/36 3
4 Anothe Coollay Pascal s Tiangle ( n n Coollay: 0( 1 0 Pascal aanged binomial coefficients as a tiangle n th ow consists of ( n fo 0, 1,... n Instucto: Işıl Dillig, CS311H: Discete Mathematics Pemutations and Combinations 19/36 Instucto: Işıl Dillig, CS311H: Discete Mathematics Pemutations and Combinations 20/36 Poof of Pascal s Identity 1 ( ( n n 1 This identity is nown as Pascal s identity Poof: ( ( n n n! 1 ( 1!(n 1! n! (!(n! Multiply fist faction by ( n 1 n 1 and second by n 1 : n! (n 1n! (!(n 1! Poof of Pascal s Identity, cont. ( n 1 Facto the numeato: ( ( n n 1 But this is exactly 1 n! (n 1n! (!(n 1! (n 1 n! (!(n 1! (n 1!! (n 1! Instucto: Işıl Dillig, CS311H: Discete Mathematics Pemutations and Combinations 21/36 Instucto: Işıl Dillig, CS311H: Discete Mathematics Pemutations and Combinations 22/36 Inteesting Facts about Pascal s Tiangle Some Fun Facts about Pascal s Tiangle, cont. What is the sum of numbes in n th ow in Pascal s tiangle (stating at n 0? Obseve: This is exactly the coollay we poved ealie! n 0 2 n Pascal s tiangle is pefectly symmetic Numbes on left ae mio image of numbes on ight Why is this the case? Instucto: Işıl Dillig, CS311H: Discete Mathematics Pemutations and Combinations 23/36 Instucto: Işıl Dillig, CS311H: Discete Mathematics Pemutations and Combinations 24/36 4
5 Pemutations with Repetitions Geneal Fomula fo Pemutations with Repetition Ealie, when we defined pemutations, we only allowed each object to be used once in the aangement But sometimes maes sense to use an object multiple times Example: How many stings of length 4 can be fomed using lettes in English alphabet? Since sting can contain same lette multiple times, we want to allow epetition! A pemutation with epetition of a set of objects is an odeed aangement of these objects, whee each object may be used moe than once P (n, denotes numbe of -pemutations with epetition fom set with n elements What is P (n,? How many ways to assign 3 jobs to 6 employees if evey employee can be given moe than one job? How many diffeent 3-digit numbes can be fomed fom 1, 2, 3, 4, 5? Instucto: Işıl Dillig, CS311H: Discete Mathematics Pemutations and Combinations 25/36 Instucto: Işıl Dillig, CS311H: Discete Mathematics Pemutations and Combinations 26/36 Combinations with Repetition Combinations help us to answe the question In how many ways can we choose objects fom n objects? Now, conside the slightly diffeent question: In how many ways can we choose objects fom n inds of objects? These questions ae quite diffeent: Fo fist question, once we pic one of the n objects, we cannot pic the same object again Fo second question, once we pic one of the n inds of objects, we can pic the same type of object again! Combination with epetition allows answeing the latte type of question! Example An ice ceam desset consists of thee scoops of ice ceam Each scoop can be one of the flavos: chocolate, vanilla, mint, lemon, aspbey In how many diffeent ways can you pic you desset? Example of combination with epetition: In how many ways can we pic 3 objects fom 5 inds of objects? Caveat: Despite looing deceptively simple, quite difficult to figue this out (at least fo me... Instucto: Işıl Dillig, CS311H: Discete Mathematics Pemutations and Combinations 27/36 Instucto: Işıl Dillig, CS311H: Discete Mathematics Pemutations and Combinations 28/36 Example, cont. Example, cont. C V M R L C V M R L Let s loo at some selections and thei epesentation: 3 scoops of chocolate: To solve poblem, imagine we have ice ceam in boxes. We stat with leftmost box, and poceed towads ight. 1 vanilla, 1 aspbey, 1 lemon: 2 mint, 1 aspbey: At evey box, you can tae 0-3 scoops, and then move to next. Denote taing a scoop by and moving to next box by Invaiant: cicles and n 1 aows (hee, 3, n 5 Ou question is equivalent to: In how many ways can we aange cicles and n 1 aows? Instucto: Işıl Dillig, CS311H: Discete Mathematics Pemutations and Combinations 29/36 Instucto: Işıl Dillig, CS311H: Discete Mathematics Pemutations and Combinations 30/36 5
6 Result Example 1 We ll denote the numbe of ways to choose objects fom n inds of objects C (n, : ( C n 1 (n, Example: In how many ways can we choose 3 scoops of ice ceam fom 5 diffeent flavos? Hee, 3 and n 5. Thus: ( 7 7! 3 3! 4! 35 Suppose thee is a bowl containing apples, oanges, and peas Thee is at least fou of each type of fuit in the bowl How many ways to select fou pieces of fuit fom this bowl? Instucto: Işıl Dillig, CS311H: Discete Mathematics Pemutations and Combinations 31/36 Instucto: Işıl Dillig, CS311H: Discete Mathematics Pemutations and Combinations 32/36 Example 2 Example 3 Conside a cash box containing $1 bills, $2 bills, $5 bills, $10 bills, $20 bills, $50 bills, and $100 bills Thee is at least five of each type of bill in the box How many ways ae thee to select 5 bills fom this cash box? Assuming x 1, x 2, x 3 ae non-negative integes, how many solutions does x 1 x 2 x 3 11 have? Instucto: Işıl Dillig, CS311H: Discete Mathematics Pemutations and Combinations 33/36 Instucto: Işıl Dillig, CS311H: Discete Mathematics Pemutations and Combinations 34/36 Example 4 Summay of Diffeent Pemuations and Combinations Suppose x 1, x 2, x 3 ae integes s.t. x 1 1, x 2 2, x 3 3. Ode mattes? Question: How many ways to pic objects fom... n objects n types of objects Then, how many solutions does x 1 x 2 x 3 11 have? Yes Pemutation P(n, n! (n! Pemutation w/ epetition P (n, n No Combination C (n, n!! (n! Combination w/ epetition C (n, (n 1!! (n 1! Instucto: Işıl Dillig, CS311H: Discete Mathematics Pemutations and Combinations 35/36 Instucto: Işıl Dillig, CS311H: Discete Mathematics Pemutations and Combinations 36/36 6
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