By David Guichard
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Extra info for An Introduction to Combinatorics and Graph Theory [Lecture notes]
Im 1 2 where the sum is over all i1 , . . , im such that i1 + · · · + im = n. 6 The Pigeonhole Principle A key step in many proofs consists of showing that two possibly different values are in fact the same. The Pigeonhole principle can sometimes help with this. 1 Pigeonhole Principle Suppose that n + 1 (or more) objects are put into n boxes. Then some box contains at least two objects. Proof. Suppose each box contains at most one object. Then the total number of objects is at most 1 + 1 + · · · + 1 = n, a contradiction.
1. Assume that the relation “friend” is symmetric. Show that if n ≥ 2, then in any group of n people there are two with the same number of friends in the group. 2. Suppose that 501 distinct integers are selected from 1 . . 1000. Show that there are distinct selected integers a and b such that a | b. Show that this is not always true if 500 integers are selected. ⇒ 3. Each of 15 red balls and 15 green balls is marked with an integer between 1 and 100 inclusive; no integer appears on more than one ball.
S(n, k)? n i n 5. Show that x = n−1 k=0 (x − k) = i=0 s(n, i)x , n ≥ 1; x is called a falling factorial. Find n−1 n n a similar identity for x = k=0 (x + k); x is a rising factorial. n n xk = xn , n ≥ 1; xk is defined in the previous exercise. The previous k k=0 exercise shows how to express the falling factorial in terms of powers of x; this exercise shows how to express the powers of x in terms of falling factorials. 6. Show that n−1 7. Prove: S(n, k) = i=k−1 8. Prove: n = k n−1 S(i, k − 1).
An Introduction to Combinatorics and Graph Theory [Lecture notes] by David Guichard