Chapter 4 Some Counting Problems; Multinomial Coefficients, The Inclusion-Exclusion Principle, Sylvester’s Formula, The Sieve Formula 4.1 Counting Permutations and Functions In this short section, we consider some simple counting problems. Let us begin with permutations. Recall that a permutation of a set, A,isanybijectionbetweenA and itself. 427 428 CHAPTER 4. SOME COUNTING PROBLEMS; MULTINOMIAL COEFFICIENTS If A is a finite set with n elements, we mentioned earlier (without proof) that A has n!permutations,wherethe factorial function, n n!(n N), is given recursively by: 7! 2 0! = 1 (n +1)!= (n +1)n!. The reader should check that the existence of the func- tion, n n!, can be justified using the Recursion Theo- rem (Theorem7! 2.5.1). Proposition 4.1.1 The number of permutations of a set of n elements is n!. Let us also count the number of functions between two finite sets. Proposition 4.1.2 If A and B are finite sets with A = m and B = n, then the set of function, BA, from| | A to B has| | nm elements. 4.1. COUNTING PERMUTATIONS AND FUNCTIONS 429 As a corollary, we determine the cardinality of a finite power set. Corollary 4.1.3 For any finite set, A, if A = n, then 2A =2n. | | | | Computing the value of the factorial function for a few inputs, say n =1, 2 ...,10, shows that it grows very fast. For example, 10! = 3, 628, 800. Is it possible to quantify how fast factorial grows com- pared to other functions, say nn or en? Remarkably, the answer is yes. A beautiful formula due to James Stirling (1692-1770) tells us that n n n! p2⇡n , ⇠ e which means that ⇣ ⌘ n! lim n =1. n p n !1 2⇡n e 430 CHAPTER 4. SOME COUNTING PROBLEMS; MULTINOMIAL COEFFICIENTS Figure 4.1: Jacques Binet, 1786-1856 Here, of course, 1 1 1 1 e =1+ + + + + + 1! 2! 3! ··· n! ··· the base of the natural logarithm. It is even possible to estimate the error. It turns out that n n n!=p2⇡n eλn, e where ⇣ ⌘ 1 1 <λ < , 12n +1 n 12n aformuladuetoJacquesBinet(1786-1856). Let us introduce some notation used for comparing the rate of growth of functions. 4.1. COUNTING PERMUTATIONS AND FUNCTIONS 431 We begin with the “Big oh” notation. Given any two functions, f : N R and g : N R,we say that f is O(g) (or f(n) is O!(g(n))) i↵there! is some N>0andaconstantc>0suchthat f(n) c g(n) , for all n N. | | | | ≥ In other words, for n large enough, f(n) is bounded by c g(n) .Wesometimeswriten>>|0toindicatethat| n is| “large.”| 1 For example λn is O(12n). By abuse of notation, we often write f(n)=O(g(n)) even though this does not make sense. The “Big omega” notation means the following: f is ⌦(g) (or f(n) is ⌦(g(n))) i↵there is some N>0anda constant c>0suchthat f(n) c g(n) , for all n N. | |≥ | | ≥ 432 CHAPTER 4. SOME COUNTING PROBLEMS; MULTINOMIAL COEFFICIENTS The reader should check that f(n)isO(g(n)) i↵ g(n)is ⌦(f(n)). We can combine O and ⌦to get the “Big theta” nota- tion: f is ⇥(g) (or f(n) is ⇥(g(n))) i↵there is some N>0andsomeconstantsc1 > 0andc2 > 0suchthat c g(n) f(n) c g(n) , for all n N. 1| || | 2| | ≥ Finally, the “Little oh” notation expresses the fact that afunction,f,hasmuchslowergrowththanafunctiong. We say that f is o(g) (or f(n) is o(g(n))) i↵ f(n) lim =0. n !1 g(n) For example, pn is o(n). 4.2. COUNTING SUBSETS OF SIZE K; MULTINOMIAL COEFFICIENTS 433 4.2 Counting Subsets of Size k; Binomial and Multi- nomial Coefficients Let us now count the number of subsets of cardinality k of a set of cardinality n,with0 k n. n Denote this number by k (say “n choose k”). Actually, in the proposition below, it will be more convenient to assume that k Z. 2 Proposition 4.2.1 For all n N and all k Z, if n 2 2 k denotes the number of subsets of cardinality k of a set of cardinality n, then 0 =1 0 ✓ ◆ n =0 if k/ 0, 1,...,n k 2{ } ✓ ◆ n n 1 n 1 = − + − (n 1, 0 k n). k k k 1 ≥ ✓ ◆ ✓ ◆ ✓ − ◆ n The numbers k are also called binomial coefficients, because they arise in the expansion of the binomial ex- pression (a + b)n,aswewillseeshortly. 434 CHAPTER 4. SOME COUNTING PROBLEMS; MULTINOMIAL COEFFICIENTS The binomial coefficients can be computed inductively using the formula n n 1 n 1 = − + − k k k 1 ✓ ◆ ✓ ◆ ✓ − ◆ (sometimes known as Pascal’s recurrence formula)by forming what is usually called Pascal’s triangle,which n is based on the recurrence for k : n n n n n n n n n n n n 0 1 2 3 4 5 6 7 8 9 10 ... 01 111 2121 31331 414641 515101051 6161520 15 61 717213535 21 7 1 818285670562881 9193684126126843691 10 1 10 45 120 210 252 210 120 45 10 1 . 4.2. COUNTING SUBSETS OF SIZE K; MULTINOMIAL COEFFICIENTS 435 Figure 4.2: Blaise Pascal, 1623-1662 n We can also give the following explicit formula for k in terms of the factorial function: Proposition 4.2.2 For all n, k N, with 0 k n, we have 2 n n! = . k k!(n k)! ✓ ◆ − Then, it is very easy to see that n n = . k n k ✓ ◆ ✓ − ◆ Remarks: (1) The binomial coefficients were already known in the twelfth century by the Indian Scholar Bhaskra. Pas- cal’s triangle was taught back in 1265 by the Persian philosopher, Nasir-Ad-Din. 436 CHAPTER 4. SOME COUNTING PROBLEMS; MULTINOMIAL COEFFICIENTS (2) The formula given in Proposition 4.2.2 suggests gen- eralizing the definition of the binomial coefficients to upper indices taking real values. Indeed, for all r R and all integers, k Z,wecan set 2 2 k r r r(r 1) (r k +1) = = − ··· − if k 0 k k! k(k 1) 2 1 ≥ ✓ ◆ ( 0if− ··· · k<0. Note that the expression in the numerator, rk,stands for the product of the k terms k terms r(r 1) (r k +1). − ··· − z }| { By convention, the value of this expression is 1 when r k =0,sothat 0 =1. 4.2. COUNTING SUBSETS OF SIZE K; MULTINOMIAL COEFFICIENTS 437 r The expression k can be viewed as a polynomial of degree k in r.Thegeneralizedbinomialcoefficients allow for a useful extension of the binomial formula (see next) to real exponents. However, beware that the symmetry identity fails when r is not a natural number and that the formula in Proposition 4.2.2 (in terms of the factorial function) only makes sense for natural numbers. We now prove the “binomial formula” (also called “bino- mial theorem”). 438 CHAPTER 4. SOME COUNTING PROBLEMS; MULTINOMIAL COEFFICIENTS Proposition 4.2.3 (Binomial Formula) For all n N and for all reals, a, b R, (or more generally, any2 two commuting variables2a, b, i.e., satisfying ab = ba), we have the formula: n n n n 1 n n 2 2 (a + b) = a + a − b + a − b + 1 2 ··· ✓ ◆ ✓ ◆ n n k k n n 1 n + a − b + + ab − + b . k ··· n 1 ✓ ◆ ✓ − ◆ The above can be written concisely as n n n n k k (a + b) = a − b . k Xk=0 ✓ ◆ Remark: The binomial formula can be generalized to the case where the exponent, r,isarealnumber(even negative). This result is usually known as the binomial theorem or Newton’s generalized binomial theorem. 4.2. COUNTING SUBSETS OF SIZE K; MULTINOMIAL COEFFICIENTS 439 Formally, the binomial theorem states that r 1 r r k k (a + b) = a − b ,rN or b/a < 1. k 2 | | Xk=0 ✓ ◆ Observe that when r is not a natural number, the right- hand side is an infinite sum and the condition b/a < 1 insures that the series converges. | | For example, when a =1andr =1/2, if we rename b as x,weget 1 1 1 (1 + x)2 = 2 xk k Xk=0 ✓ ◆ 1 1 1 1 1 =1+ 1 k +1 xk k! 2 2 − ··· 2 − Xk=1 ✓ ◆ ✓ ◆ 1 k 1 1 3 5 (2k 3) k =1+ ( 1) − · · ··· − x − 2 4 6 2k k=1 · · ··· X k 1 1 ( 1) − (2k)! =1+ − xk, 22k(2k 1)(k!)2 k=1 − X k 1 1 ( 1) − 2k =1+ − xk 22k(2k 1) k k=1 − ✓ ◆ X k 1 1 ( 1) − 1 2k 2 =1+ − − xk, 22k k k 1 Xk=1 ✓ − ◆ 440 CHAPTER 4. SOME COUNTING PROBLEMS; MULTINOMIAL COEFFICIENTS which converges if x < 1. | | The first few terms of this series are 1 1 1 1 5 (1 + x)2 =1+ x x2 + x3 x4 + 2 − 8 16 − 128 ··· For r = 1, we get the familiar geometric series − 1 =1 x + x2 x3 + +( 1)kxk + , 1+x − − ··· − ··· which converges if x < 1. | | Remark: The numbers, 1 2n C = , n n +1 n ✓ ◆ are the Catalan numbers.Theyarethesolutionofmany counting problems in combinatorics. Proposition 4.2.4 The number of injections between a set, A, with m elements and a set, B, with n ele- ments, where m n, is given by n! (n m)! = n(n 1) (n m +1). − − ··· − 4.2. COUNTING SUBSETS OF SIZE K; MULTINOMIAL COEFFICIENTS 441 Counting the number of surjections between a set with n elements and a set with p elements, where n p,is harder.