Question 5: Spinal Tap has finally gotten the band back together for a photo op. The band consists of David, Nigel, Derek, and their latest drummer (whose name no one bothered to learn). There are n3 people who wish to have their picture taken with members of the band (so with the band members there are n+1 people total, where n7 ). The drummer appears in every picture. David, Nigel, and Derek hate each other and refuse to be in any picture together. Let k be an integer such that 3kn3. Each picture has exactly k+1 people in it consisting of - the drummer and k fans, or - the drummer, exactly one of David, Nigel, or Derek, and k1 fans. A picture is unique if it contains a unique set of people. a) Use the Sum Rule to count the number of unique pictures. b) Use the Complement Rule to count the number of unique pictures. c) Use the above two results to prove that (nk)=(n3k3)+3(n3k2)+3(n3k1)+(n3k). d) David, Nigel, and Derek finally agree to be in one picture together, but refuse to stand next to each other. How many ways can David, Nigel, Derek and n3 fans be arranged on a line such that David, Nigel, and Derek are not standing next to one another? (The drummer died mysteriously and so does not appear in the picture.)