By David Guichard

**Read Online or Download An Introduction to Combinatorics and Graph Theory [Lecture notes] PDF**

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**Extra resources for An Introduction to Combinatorics and Graph Theory [Lecture notes]**

**Example text**

A52 ) are integers, not necessarily distinct. Show that there are two, ai and aj with i = j, such that either ai + aj or ai − aj is divisible by 100. Show that this is not necessarily true for integers (a1 , a2 , . . , a51 ). 5. Suppose five points are chosen from a square whose sides are length s. ) Show that √ two of the points are at most s 2/2 apart. Find five points so that no two are less than s 2/2 apart. 6. Show that if the edges of a K6 are colored with two colors, there are at least two monochromatic triangles.

N + i − 1)! = (−1)i i! (n − 1)! n+i−1 n+i−1 = (−1)i = (−1)i . i n−1 Thus ∞ ∞ n+i−1 −n i n+i−1 i (x + 1) = (−1) x = (−x)i . n−1 n−1 i=0 i=0 Now replacing x by −x gives −n (1 − x) ∞ = i=0 −n So (1 − x) is the generating function for 1, ∞ · 2, . . , ∞ · n} of size i. n+i−1 i x. 1 Newton’s Binomial Theorem 53 In many cases it is possible to directly construct the generating function whose coefficients solve a counting problem. 3 Find the number of solutions to x1 + x2 + x3 + x4 = 17, where 0 ≤ x1 ≤ 2, 0 ≤ x2 ≤ 5, 0 ≤ x3 ≤ 5, 2 ≤ x4 ≤ 6.

N−1 k Qn = n! k=0 (−1)k−1 k=1 Note that the limits on the two sums are not identical. 1 . (k − 1)! 2. 1. Prove that Dn = nDn−1 + (−1)n when n ≥ 2. 2. Prove that Dn is even if and only if n is odd. Qn ? n→∞ n! 4. Find the number of permutations of 1, 2, . . , 8 that have no odd number in the correct position. 3. 1. What is lim 5. Find the number of permutations of 1, 2, . . , 8 that have at least one odd number in the correct position. 6. How many permutations of [n] have exactly k numbers in their correct positions?