Let an equal the number of strings of length n made up of 0,1,2 that contain...

 

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Let an equal the number of strings of length n made up of 0,1,2 that contain no two consecutive O's. (a) Find and verify a recurrence relation for an, and give initial conditions. (You do not need to solve this recurrence.) (b) Find a generating function for an. For any positive integer n, give a combinatorial proof of n ("+2) = (3) + 2(2)+(4) 3 You receive at most 10 points if proving it by other methods. Paragraph BI UA + v Apply the Inclusion-Exclusion Principle to find the number of integral solutions of x1 x2 x3 + x4 = 15 such that -1 1x14, 1 < x2 < 5,0x35, -2x4- 5,2x4. Paragraph BI UV MEYE + (a) (12 points) How many of (1, 2, ..., 10} must be selected in order to guarantee two of the chosen integers add up to 10? Prove your claim and explain why your answer is the smallest possible. (b) (8 points) There are 25 graduate students in the math department of a university. Each of them must take two computer science courses out of four available courses. Show that at least 5 students take exactly the same two computer science courses. Let an equal the number of strings of length n made up of 0,1,2 that contain no two consecutive O's. (a) Find and verify a recurrence relation for an, and give initial conditions. (You do not need to solve this recurrence.) (b) Find a generating function for an. For any positive integer n, give a combinatorial proof of n ("+2) = (3) + 2(2)+(4) 3 You receive at most 10 points if proving it by other methods. Paragraph BI UA + v Apply the Inclusion-Exclusion Principle to find the number of integral solutions of x1 x2 x3 + x4 = 15 such that -1 1x14, 1 < x2 < 5,0x35, -2x4- 5,2x4. Paragraph BI UV MEYE + (a) (12 points) How many of (1, 2, ..., 10} must be selected in order to guarantee two of the chosen integers add up to 10? Prove your claim and explain why your answer is the smallest possible. (b) (8 points) There are 25 graduate students in the math department of a university. Each of them must take two computer science courses out of four available courses. Show that at least 5 students take exactly the same two computer science courses.