4 4 Permutations and Combinations of multisets Multisets

4. 4 Permutations and Combinations of multisets § Multisets : A multiset is a set in which an item may appear more than once. . § Sets and multisets are quite similar: both support the basic set operations of union, intersection, and difference. § item ai ni， § {n 1 • a 1, n 2 • a 2, …, nk • ak} § Example：{a, a, b, b, c} § {4 • a, 2 • b, 1 • c} § { • a 1, • a 2, …, • ak}

4. 4 Permutations and Combinations of multisets § 4. 4. 1 Permutations of multisets § If S is a multiset, a r-permutation of S is an ordered arrangement of r of the objects of S. If the total number of objects of S is n，then a n-permutation of S will also be called a permutation of S. § For example, if S={2 • a, 1 • b, 3 • c}, then § aacb acbc cacc are 4 -permutations of S, § “abccac” is a permutation of S. § The multiset S has no 7 -permutations since 7>2+1+3=6, the number of objects of S.

§ Theorem 4. 10: Let S be a multiset with objects of k different types where each has an infinite repetition number. Then the number of r-permutations of S is kr. § Proof: In constructing a r-permutation of S, § we can choose the first item can be an object of any one of the k types. § Similarly the second item to be an object of any one of the k types, and so on. § Since all repetition numbers of S are infinite, the number of different choices for any item is always k and does not depend on the choices of any previous items. § By the multiplication principle, the r items can be chose in kr ways.

§ Corollary 4. 4: Let S={n 1 • a 1, n 2 • a 2, …, nk • ak}， and ni r for each i=1, 2, …, n，then the number of r-permutations of S is kr.

§ Theorem 4. 11: Let multiset S={n 1 • a 1, n 2 • a 2, …, nk • ak}, and n=n 1+n 2+…+nk=|S|. Then the number of permutations of S equals n!/(n 1!n 2!…nk!)。 § Proof: We can think of it this way. There are n places, and we want to put exactly one of the objects of S in each of the places. § Since there are n 1 a 1’s in S, we must choose a subset of n 1 places from the set of n places. § C(n, n 1) § We next decided which places are to be occupied by the a 2’.

§ Example What is the number of permutations of the letters in the word Mississippi?

§ Let S={n 1 • a 1, n 2 • a 2, …, nk • ak}, and n=n 1+n 2+…+nk=|S|，then the number N of rpermutations of S equals § (1)0 r>n § (2)n!/(n 1!n 2!…nk!) r=n § (3)kr ni r for each i=1, 2, …, n § (4)If r<n, there is, in general, no simple formula for the number of r-permutations of S. § Nonetheless a solution can be obtained by the technique of generating functions, and we discuss this in 4. 6.

§ 4. 4. 2 Combinations of multisets § If S is a multiset, a r-combination of S is an unordered selection of r of the objects of S. § Thus an r-combination of S is a submultiset of S. If S has n objects，then there is only one n-combination of S, namely, S itself. § If S contains objects of k different types, then there are k 1 -combinations of S. § Example If S={2 • a, 1 • b, 3 • c}, then the 3 -combinations of S are § {2 • a, 1 • b}, {2 • a, 1 • c}, {1 • a, 1 • b, 1 • c}, {1 • a, 2 • c}, {1 • b, 2 • c}, {3 • c}.

§ Theorem 4. 12: Let S ={ ·a 1, ·a 2, …, ·ak}. Then the number of r-combinations of S equals C(k+r-1, r)。 § Proof. (1)The number of r-combinations of S equals the number of solutions of the equation § where x 1, x 2, …, xk are non-negative integers (2)We show that the number of these solutions equals the number of permutations of the multiset T={(k-1)· 0, r· 1}.

§ Corollary 4. 5: Let S={n 1 • a 1, n 2 • a 2, …, nk • ak}， and ni r for each i=1, 2, …, n. Then the number of r-combinations of S is C(k+r-1, r). § Example Suppose that a cookie shop has seven different kinds of cookies. How many different ways can twelve cookies be chosen?

§ Example How many integer solutions does the equation x 1+x 2+x 3=11 have, where x 1, x 2, and x 3 are non-negative integers? § Solution To count the number of solutions, we note that a solution corresponds to a way of selecting 11 items from a set with three elements.

§ Example How many solutions are there to the equation x 1+x 2+x 3=11 where x 1 3, x 2 1, and x 3 0? § Solution We introduce new variables: § y 1=x 1 -3, y 2=x 2 -1, y 3=x 3, § and the equation becomes y 1+y 2+y 3=7 where y 1, y 2, and y 3 are non-negative integers

§ Corollary 4. 6 Let S={ ·a 1, ·a 2, …, ·ak}, and r k. Then the number of r-combinations of S so that each of the k types of objects occurs at least once equals C(r-1, k-1). § Proof: r-combinations of S so that each of the k types of objects occurs at least once, a 1, a 2, …, ak， § r-k combinations, +a 1, a 2, …, ak，r-combinations of S so that each of the k types of objects occurs at least once

§ Let S={n 1 • a 1, n 2 • a 2, …, nk • ak}, and n=n 1+n 2+…+nk=|S|， then the number N of r-combinations of S equals § (1) 0 r>n § (2) 1 r=n § (3) N=C(k+r-1, r) ni r for each i=1, 2, …, n. § (4) If r<n, and there is, in general, no simple formula for the number of r-combinations of S. Nonetheless a solution can be obtained by the inclusion-exclusion principle and technique of generating functions, and we discuss these in 4. 5 and 4. 6.

4. 5 Inclusion-Exclusion principle and Applications § 4. 5. 1 Inclusion-Exclusion principle § Theorem 4. 13：Let A and B be finite sets. Then |A∪B|=|A|+|B|-|A∩B|。 § Proof：Because A∪B=A∪(B-A)，and A∩(BA)= , by theorem we obtain |A∪B|=|A|+|B-A|. § |B-A|=？ § Theorem 4. 14：Let A 1, A 2, …, An be finite sets. Then

§ Corollary 4. 7: Let S be a finite set, and P 1, P 2, …, Pn be n properties referring to the objects in S. Let § Ai={x|x S and x has property Pi}(i=1, 2, …, n) be the subset of S which have property Pi (and possibly other properties). Then the number of objects of S which have none of the properties P 1, P 2, …, Pn is given by

§ Example: In a certain school, the pupils have to study France, German, or English. Each pupil must study at least one of the three. Among a group of 100 pupils, 42 are studying France, 45 are studying German, 65 are studying English, 15 are studying France and German, 20 are studying France and English, 25 are studying German and English. Find the number of pupils who are studying all the three subjects. And Find the number of pupil who are studying English only. § A, France § B, German § C, English § the number of pupil who are studying English only § |C|-|A∩C|-|B∩C|+|A∩B∩C|=28

§ Example：How many hex strings of length r contain 0, 1, and 2? § Solution：Let S be the set of length r’s hex strings. § Let A be the set of length r’s hex strings which do not contain 0 § Let B be the set of length r’s hex strings which do not contain 1 § Let C be the set of length r’s hex strings which do not contain 2.

§ Example：Find the number of integers between 1 to 1000, inclusive, which are divisible by none of 5, 6, and 8. § Solution: Let S be the set consisting of the first thousand positive integers. § Let P 1 be the property that an integer is divisible by 5. § Let P 2 be the property that an integer is divisible by 6. § Let P 3 be the property that an integer is divisible by 8. § For i=1, 2, 3 let Ai be the set consisting of those integers in S with property Pi.

§ Integers in the set A 1∩A 2 are divisible by both 5 and 6(lcm{5, 6}=30). Integers in the set A 1∩A 3 are divisible by both 5 and 8(lcm{5, 8}=40). Integers in the set A 2∩A 3 are divisible by both 6 and 8(lcm{6, 8}=24). § Integers in the set A 1∩A 2∩A 3 are divisible by 5 , 6 and 8(lcm{5, 6, 8}=120). § By the inclusion-exclusion principle, the number of integers between 1 to 1000 that are divisible by none of 5, 6, and 8 equals

§ Next: Applications of Inclusion-Exclusion principle, Generating functions, § Exercise P 96 22, 23 (Sixth) OR P 83 22, 33(Fifth); § P 99 10, 12, 14, 16, 18, 20(Sixth) OR P 86 10, 12, 14, 16, 18, 20(Fifth) § 1. Prove Theorem 4. 9 (4)(7). § 2. How many solutions are there to the equation x 1+x 2+x 3+x 4+x 5=25, where x 1, x 2, x 3, x 4, and x 5 are nonnegative integers? § 3. How many strings of 20 decimal digits are there that contain two 0’s, four 1’s, three 2’s, one 3, two 4’s, three 5’s, two 7’s, and three 9’s? § 4. How many positive integers less than 1, 000 have the sum of their digits equal to 19? § 5. A football team of 11 players is to be selected from a set of 16 players, 5 of whom can only play in the backfield, 9 of whom can only play on the line, and 2 of whom can play either in the backfield or on the line. Assuming a football team has 7 men on the line and 4 in the backfield, determine the number of football teams possible. § 6. Find the number of integers between 1 and 10, 000 inclusive which are not divisible by 4, 5, or 6.