By Russell Merris

A mathematical gem–freshly wiped clean and polished

This e-book is meant for use because the textual content for a primary path in combinatorics. the textual content has been formed through pursuits, particularly, to make complicated arithmetic available to scholars with quite a lot of talents, pursuits, and motivations; and to create a pedagogical software, important to the wide spectrum of teachers who carry numerous views and expectancies to the sort of course.

Features retained from the 1st edition:

Lively and fascinating writing style

Timely and acceptable examples

Numerous well-chosen exercises

Flexible modular format

Optional sections and appendices

Highlights of moment version enhancements:

Smoothed and polished exposition, with a sharpened specialise in key ideas

Expanded dialogue of linear codes

New non-compulsory part on algorithms

Greatly elevated tricks and solutions section

Many new workouts and examples

**Reviews:**

“…broad and interesting…” (Zentralblatt Math, Vol.1035, No.10, 2004)

“...engagingly written...a powerful studying tool...” (American Mathematical per month, March 2004)

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**Example text**

How many different lattice paths terminate at ðr; sÞ? , such that xk ! yk for each point Pk ¼ ðxk ; yk Þ. Show that (a) c1 ¼ 1; c2 ¼ 2, and c3 ¼ 5. P (b) cnþ1 ¼ nr¼0 cr cnÀr . (Hint: Lattice paths ‘‘touch’’ the line y ¼ x for the last time at the point ðn; nÞ. Count those whose next-to-last touch is at the point ðr; rÞ). (c) cn is the nth Catalan number of Exercises 13–14, n ! 1. 17 Let X and Y be disjoint sets containing n and m elements, respectively. In how many different ways can an ðr þ sÞ-element subset Z be chosen from X [ Y if r of its elements must come from X and s of them from Y?

Exercises 17 (a) using algebraic arguments. (b) using combinatorial arguments. 10 Suppose n, k, and r are integers that satisfy n ! k ! r ! 0 and k > 0. Prove that (a) Cðn; kÞCðk; rÞ ¼ Cðn; rÞCðn À r; k À rÞ. 11 (b) Cðn; kÞCðk; rÞ ¼ Cðn; k À rÞCðn À k þ r; rÞ. Pn nÀr (c) . j ¼ 0 Cðn; jÞCð j; rÞ ¼ Cðn; rÞ2 Pn jþk Cðn; jÞ ¼ Cðn À 1; k À 1Þ. (d) j ¼ k ðÀ1Þ Â Pn Ã2 P2n Prove that r ¼ 0 Cðn; rÞ ¼ s ¼ 0 Cð2n; sÞ. 12 Prove that Cð2n; nÞ, n > 0, is always even. 13 Probably first studied by Leonhard Euler (1707–1783), the Catalan sequence* 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862; .

1485–1567), among the most celebrated algebraists of the sixteenth century. Also known for numerological prophesy, Stifel predicted publicly that the world would end on October 3, 1533. 44 The Mathematics of Choice Á À Proof. ;rk is the number of n-letter ‘‘words’’ that can be assembled using r1 copies of one ‘‘letter’’, say A1 ; r2 copies of a second, A2 ; and so on, finally using rk copies of some kth character, Ak . The theorem is proved by counting these words another way and setting the two (different-looking) answers equal to each other.