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Question 2: Recall that K, where n 2 is an integer, denotes the complete graph with n vertices. This graph has ) edges, i.e., one edge for each pair of distinct vertices. Let k > 1 be an integer and let L be a set of size k. If each edge of K. is labeled with one element of L, then we say that K, is k-labeled. Three pairwise distinct vertices u, v, and w of K, define a boring triangle, if the three edges wv, ww, and vw have the same label In this question, you will prove the following statement Sin,k) for all integers n and k with n 3.k! S(n,): Every k-labeled K, has a boring triangle. (Note that in class, we have proved statement S(n. 2) for all n > 6.) The proof will be by induction on Base case: Explain, in a few sentences, why statement Sin, 1) is true for all integers 1 2 3 Induction step: Let k > 2 and assume that statement S(m.k-1) is true for all integers m2 3. (k-1). Prove that statement Sin,k) is true for all integers n 23. kl. Hint: Let u be an arbitrary vertex of K. Argue that among all edges that have u as a vertex, at least f(n-1)/41 have the same label Question 3: Let k 2 be an integer and consider the set S = {1,2....,k). For any integer n> 2, determine the mumber of sequences (1,2,.....) of lengthn, where each element belongs to S and no two consecutive elements are the same. Question 4: Zoltan's House of Pizzn is a popular restaurant in downtown Ottawa. The daily dinner special is Zoltan's Meal Deal: You choose between standard dough and gluten free dough: you choose one of chicken, beef, pork, and shrimps you choose one out of five different sauces; you choose three out of seven different cheeses; you choose any subset of twenty different toppings, and finally, you choose four out of nine different beers. Determine the number of Zoltan's Meal Deals, if the cheeses chosen must be distinct and the beers chosen must be distinct. Determine the number of Zoltan's Meal Deals, if the cheese chosen are not necessarily distinct and the beers chosen are not necessarily distinct. Question 5: This fall term, 230 students have registered for COMP 2804B. While watching video lectures, 150 students do not wear pants and 110 students drink beer. Determine the best possible lower bound on the number of students who do not wear pants and drink beer while watching video lectures, Question 2: Recall that K, where n 2 is an integer, denotes the complete graph with n vertices. This graph has ) edges, i.e., one edge for each pair of distinct vertices. Let k > 1 be an integer and let L be a set of size k. If each edge of K. is labeled with one element of L, then we say that K, is k-labeled. Three pairwise distinct vertices u, v, and w of K, define a boring triangle, if the three edges wv, ww, and vw have the same label In this question, you will prove the following statement Sin,k) for all integers n and k with n 3.k! S(n,): Every k-labeled K, has a boring triangle. (Note that in class, we have proved statement S(n. 2) for all n > 6.) The proof will be by induction on Base case: Explain, in a few sentences, why statement Sin, 1) is true for all integers 1 2 3 Induction step: Let k > 2 and assume that statement S(m.k-1) is true for all integers m2 3. (k-1). Prove that statement Sin,k) is true for all integers n 23. kl. Hint: Let u be an arbitrary vertex of K. Argue that among all edges that have u as a vertex, at least f(n-1)/41 have the same label Question 3: Let k 2 be an integer and consider the set S = {1,2....,k). For any integer n> 2, determine the mumber of sequences (1,2,.....) of lengthn, where each element belongs to S and no two consecutive elements are the same. Question 4: Zoltan's House of Pizzn is a popular restaurant in downtown Ottawa. The daily dinner special is Zoltan's Meal Deal: You choose between standard dough and gluten free dough: you choose one of chicken, beef, pork, and shrimps you choose one out of five different sauces; you choose three out of seven different cheeses; you choose any subset of twenty different toppings, and finally, you choose four out of nine different beers. Determine the number of Zoltan's Meal Deals, if the cheeses chosen must be distinct and the beers chosen must be distinct. Determine the number of Zoltan's Meal Deals, if the cheese chosen are not necessarily distinct and the beers chosen are not necessarily distinct. Question 5: This fall term, 230 students have registered for COMP 2804B. While watching video lectures, 150 students do not wear pants and 110 students drink beer. Determine the best possible lower bound on the number of students who do not wear pants and drink beer while watching video lectures