Chapter 7 COMBINATIONS AND PERMUTATIONS. where we have the specific formula for the binomial coefficients:

Size: px

Start display at page:

Download "Chapter 7 COMBINATIONS AND PERMUTATIONS. where we have the specific formula for the binomial coefficients:"

Adrian Harvey
5 years ago
Views:

1 Chapter 7 COMBINATIONS AND PERMUTATIONS We have see i the previous chapter that (a + b) ca be writte as 0 a % a & b%þ% a & b %þ% b where we have the specific formula for the biomial coefficiets: '!!(&)! ' We ow loo at a differet iterpretatio of these umbers ad will see why called " choose." Let M = ( + x )( + x 2 )( + x ). Expadig this product, we get M = + x + x 2 + x + x x 2 + x x + x 2 x + x x 2 x. is The terms of this expasio correspod to the subsets of S = {x, x 2, x }. That is, we ca associate the term with the empty subset of S; the terms x, x 2, ad x with the sigleto subsets of S; the terms x x 2, x x, ad x 2 x with the doubleto subsets; ad x x 2 x with S itself. (S is the oly subset with elemets.) Next we replace each of x, x 2, ad x by x i our two expressios for M. This results i ( + x) = + x + x 2 + x. Thus we see that for = 0,, 2, is the umber of ways of choosig a subset of elemets from a set S of elemets. Similarly, oe ca see that the umber of ways of choosig elemets from a set of elemets is. 5

2 Sice For example, the set {, 2,, 4, 5} with 5 elemets has 5 ' ' 0, 5 subsets havig elemets. it is ot too difficult to write out all te of these subsets as {, 2, }, {, 2, 4}, {, 2, 5}, {,, 4}, {,, 5}, {, 4, 5}, {2,, 4}, {2,, 5}, {2, 4, 5}, {, 4, 5}. If we drop the braces eclosig the elemets of each subset, the resultig sequece is said to be a combiatio of thigs chose from the set {, 2,, 4, 5}. Thus the te combiatios of thigs chose from this set of 5 objects are, 2, ;, 2, 4;, 2, 5;,, 4;,, 5;, 4, 5; 2,, 4; 2,, 5; 2, 4, 5;, 4, 5. Note that chagig the order i which the objects of a combiatio are writte does ot chage the combiatio. For example,, 2, 4 is the same combiatio as, 4, 2. The formula ' & tells us that the etries o row of the Pascal Triagle read the same left to right as they do right to left. The combiatorial sigificace of this formula is that the umber of ways of choosig elemets from a set of elemets is equal to the umber of ways of omittig - of the elemets. A problem aalogous to that of fidig the coefficiets of a biomial expasio is that of fidig the coefficets i (x + y + z). These coefficiets, called the triomial coefficiets, are aturally more complicated but, fortuately, ca be expressed i terms of the biomial coefficiets i the followig way. Let us loo for the coefficiet of x 6 y z i (x + y + z) 0. From this product of te factors we must choose six x s, three y s, ad oe z. We ca choose six x s from a set of te i 0 6 ways ad three 52

3 y s from the remaiig four factors i 4 ways, ad the z from the remaiig factor i way. Therefore the triomial coefficiet of x 6 y z i (x + y + z) 0 ca be writte as or, sice as We ca obtai a alterate, 0 4 ', 6. represetatio of this umber, however, by choosig the z first i 0 ways, the the six x s from the remaiig ie, ad fially the three y s from the remaiig three. Thus the coefficiet would appear as or Hece By ' choosig the y s first, oe ca see that this coefficiet could also be expressed as or 0 7. The reader may fid other forms of the coefficiet. This problem ca be geeralized similarly to fid the coefficiets of (x%y%z%þ%w) ; they are called multiomial coefficiets. It ca readily be show that the coefficiet of x a y b z c ÿw d i the expasio of (x%y%z%þ%w) is! a!b!c!ÿd! where, of course, the sum a%b%c%þ%d of the expoets must be. Aother iterestig problem, ad oe with frequet applicatios, is that of fidig the umber of ways i which oe ca arrage a set of objects i a row, that is, the umber of permutatios of the set. Let us cosider the set of four objects a, b, c, d. They ca be arraged i the followig ways: a b c d b a c d c a b d d a b c a b d c b a d c c a d b d a c b a c b d b c a d c b a d d b a c a c d b b c d a c b d a d b c a a d b c b d a c c d a b d c a b a d c b b d c a c d b a d c b a 5

4 Rather tha write them all out, if we are oly iterested i the umber of arragemets, we may thi of the problem thus: We have four spaces to fill. If we put, for example, the b i the first, we have oly the a, c, ad d to choose from i fillig the remaiig three. Ad if we put the d i the secod, we have oly a ad c for the remaiig; ad so forth. So we have four choices for the first space, three for the secod, two for the third, ad oe for the fourth. This gives us 4@@2@, or 4! arragemets of four objects. This argumet ca be used to show that there are! arragemets of objects. We may also cosider the possibility of arragig, i a row, r objects chose from a set of. We have choices for the first space, - for the secod, - 2 for the third, ad so o. Fially we have - r + choices for the rth space, givig a total of ( - )( - 2)ÿ( - r + ) possible arragemets (or permutatios). This ca be writte i terms of factorials as follows: (&)(&2)ÿ(&r%) ' (&)(&2)ÿ(&r%)(&r)(&r&)ÿ@2@ (&r)(&r&)ÿ@2@ '! (&r)!. It should be oted that this is ot the umber of combiatios of r objects tae from a set of, sice i permutatios order is importat; i combiatios it is ot. For example, if we cosider the three objects a, b, ad c, the umber of permutatios of two objects chose from them the arragemets ab, ba, bc, cb, ca, ac. However, the umber of combiatios is ad the combiatios are a ad b, b ad c, ad a ad c.! 2!! ', Next we defie eve ad odd permutatios of, 2,..., ; this topic is used i Chapter 9 ad i higher algebra. We begi with the case =, that is the umbers, 2,. With each permutatio i, j, of these three umbers, we associate the product of differeces p = (j - i)( - i)( - j). If p is positive, the associated permutatio is called eve; if p is egative, the associated permutatio is odd. Three of the! permutatios of, 2, are eve ad three are odd. The eve oes are listed i the first colum, ad the odd oes i the secod colum: 54

5 , 2,,, 2 2,, 2,,,, 2, 2, For geeral, a permutatio i, j, h,,..., r, s of, 2,,..., is associated with the product p = [(j - i)][(h - i)(h - j)][( - i)( - j)( - h)]... [(s - i)(s - j)(s - h)(s - )...(s - r)] of all the differeces of two of i, j, h,,..., r, s i which the umber that appears first is subtracted from the other. If the permutatio i, j, h,,..., r, s is writte i the otatio a, a 2, a, a 4,..., a -, a, the the product p taes the form p ' [(a 2 & a )][(a & a )(a & a 2 )][(a 4 & a )(a 4 & a 2 )(a 4 & a )]ÿ [(a & a )(a & a 2 )(a & a )(a & a 4 )ÿ(a & a & )]. If the product p is positive, the permutatio is eve; if p is egative, the permutatio is odd. Problems for Chapter 7. Write out all the combiatios of two letters chose from a, b, c, d, ad e. 2. Write out all the combiatios of three letters chose from a, b, c, d, ad e.. Write out all the permutatios of two letters chose from a, b, c, d, ad e. 4. Write out all the permutatios of three letters chose from a, b, c, d, ad e. 5. Fid the positive iteger that is the coefficiet of x y 7 z 2 i (x + y + z) Express the triomial coefficiet of the previous problem i six ways as a product of two biomial coefficiets. 7. How may combiatios are there of, 2,, or 4 elemets from a set of 5 elemets? 8. How may o-empty proper subsets are there of a set of elemets? That is, how may combiatios are there of, 2,..., or - elemets? 9. Express the coefficiet of x y 7 w 2 i (x + y + z + w) 2 i six differet ways as a product of two biomial coefficiets. 55

6 0. Express the coefficiet of x 2 y z 4 w 2 i (x + y + z + w) i six differet ways as a product of three biomial coefficiets.. Fid the coefficiet of x 2 y 9 z w i (2x + y - z + w) Fid the coefficiet of x r yzw i (x + y + z + w) r+.. Show that x 5 y 2 z 9 has the same coefficiet i (x + y + z + w) 6 as i (x + y + z) What is the relatio betwee the coefficiet of xy 7 z 2 i (x + y - z) 0 ad its coefficiet i (x - y + z + w) 0? 5. Let a, b, ad be positive itegers, with > a + b. Show that a &a b % b &b a& % a &a b& ' % a &a% b. 6. Express the coefficiet of x 2 y 4 z 6 i (x + y + z) 2 as the sum of three of the triomial coefficiets i the expasio of (x + y + z). 7. What is the sum of all the triomial coefficiets i (x + y + z) 00? 8. What is the sum of the coefficiets i each of the followig: (a) (x + y - z) 00? (b) (x - y + z - w) 00? 9. List the eve permutatios of, 2,, List the odd permutatios of, 2,, 4. R 2. Let P be a permutatio i, j, h,... of, 2,,...,. (a) Show that if i ad j are iterchaged, P chages from odd to eve or from eve to odd. (b) Show that if ay two adjacet terms i P are iterchaged, P chages from odd to eve or from eve to odd. (c) Show that the iterchage of ay two terms i P ca be cosidered to be the result of a odd umber of iterchages of adjacet terms. (d) Show that if ay two terms i the permutatio P are iterchaged, P chages from odd to eve or from eve to odd. 56

7 (e) Give that $ 2, show that half of the permutatios of, 2,..., are eve ad half! are odd, that is, that there are eve permutatios ad the same umber of odd 2 oes. R 22. (a) Let P be a permutatio a, b, c, d of the umbers, 2,, 4. Let d = 4 ad let Q be the associated permutatio a, b, c of, 2,. Show that P ad Q are either both eve or both odd. (b) Let R be a permutatio i, j,..., h, of the umbers, 2,..., -, i which the last term of R is. Let S be the associated permutatio i, j,..., h of, 2,..., - obtaied by droppig the last term of R. Show that R ad S are either both eve or both odd. 2. How may triples of positive itegers r, s, ad t are there with r < s < t ad: (a) r + s + t = 52? (b) r + s + t = 52? The arragemet has the property that the umbers icrease as oe goes dow or to the right. (a) How may other arragemets are there of the umbers, 2,..., 8 i 2 rows ad 4 colums with this property? (b) How may arragemets are there of the umbers, 2,..., 4 i 2 rows ad 7 colums with this property? (c) How may arragemets are there of the umbers, 2,..., 2 i rows ad 4 colums with this property? 57

Lecture Overview. 2 Permutations and Combinations. n(n 1) (n (k 1)) = n(n 1) (n k + 1) =

Lecture Overview. 2 Permutations and Combinations. n(n 1) (n (k 1)) = n(n 1) (n k + 1) = COMPSCI 230: Discrete Mathematics for Computer Sciece April 8, 2019 Lecturer: Debmalya Paigrahi Lecture 22 Scribe: Kevi Su 1 Overview I this lecture, we begi studyig the fudametals of coutig discrete objects.