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- 1. Given 10 numbers a1, a2,..., a10, from the set S of natural numbers S = {1,2,3,..., 100}. Prove that there exist two (or more) subsets of the 10 numbers whose elements sum to the same value. Food for thought - how would you determine that 10 values are needed? 2. Given the set of S natural numbers S= {1, 2, 3,..., 100} select fifty one numbers from S. Prove that at least one of the numbers you chose is a multiple of another number that you chose. 3. Evan is training for baseball season. His coach wants him to pitch or hit 90 times over the course of the 53 days before the season starts. He plans to do at least one pitching or hitting session a day. Prove that there exists a span of consecutive days that Evan will complete exactly 15 sessions. 4. Use the following Venn diagram for this question. 3 (a) List the regions of the Venn diagram that make the following statement true: (-p→q) → r. (b) Construct an equivalent statement to the one in (a) using only A, V, and/or using as few…Suppose you have a set of 15 elements, and you want to form subsets from this set. How many different subsets can you create, including the empty set and the set itself? (A) 15 (B) 30 (C) 225 (D) 256A drawer contains 10 red socks, 10 blue socks, and 10 green socks. What is the minimum number of socks you need to take out of the drawer in order to guarantee you have at least 5 socks of the same color? Enter your answer as an integer (i.e., do not use words):
- A bag contains four red marbles, two green ones, one lavender one, three yellows, and three orange marbles. HINT [See Example 7.] How many sets of four marbles include none of the red ones? setsDiscrete MathHow many license plates can be made using either two uppercase English letters followed by four digits or two digits followed by four uppercase English letters? For the toolbar, press ALT+F10 (PC) or ALT+FN+F10 (Mac). BIUS Paragraph Arial 10pt A v I ... O WORDS POWERED BY T How many numbers must be selected from the set {1, 2, 3, 4, 5, 6} to guarantee that at least one pair of these numbers add up to 7? For the toolbar, press ALT+F10 (PC) or ALT+FN+F10 (Mac). BIUS Paragraph Arial 10pt A ...