ICS 6D Due: Wednesday, November 26, 2014, 3:00PM Homework 7 Instructor: Sandy Irani Leave your answer for the questions below as an arithmetic expression, including the P (n, k) or notation. You do not have to compute a final numeric value. n k 1. 20 identical printing jobs are sent to 5 different printers. (a) How many ways are there to distribute the jobs? (b) One of the printers is much faster than the other four. How many ways are there to distribute the jobs if the faster printer gets at least 7 jobs? (c) One of the printers is almost out paper. How many ways are there to distribute the jobs so that the printer that is almost out of paper does not get more than three jobs? (In addition to the fact that the faster one gets at least 7 jobs). 2. A man is distributing his coin collection with 35 coins to his five grandchildren. How many ways are there to distribute the coins if: (a) The coins are all the same. (b) The coins are all distinct. (c) The coins are the same and each grandchild gets the same number of coins. (d) The coins are all distinct and each grandchild gets the same number of coins. 3. An employee of a grocery store is placing an order for soda. There are 8 varieties of soda and they are sold in cases. Each case contains is all the same variety. The store will order 50 cases total. (a) How many ways are there to place the order? (b) How many ways are there to place te order if she orders at least 3 of each variety? (c) How many ways are there to place the order if she does not order more than 20 cases of Coke? (d) How many ways are there to place the order if she does not order more than 25 of any single variety? (e) How many ways are there to place the order if she does not order more than 20 of any single variety? 4. Consider product (x + y + z)2 . If the expression is multiplied out and like terms collected, the result is: x2 + y 2 + z 2 + 2xy + 2yz + 2xz. Suppose instead, we do the same to the product: (v + w + x + y + z)25 . (a) What is the coefficient of the term v 9 w2 x5 y 7 z 2 ? (b) How many different terms are there? (Two terms are the same if the degree of each variable is the same). 5. How many paermutations are there of each of the following multi-sets. (A multi-set can have more than one occurrence of the same time). 2 (a) S = {N, U, M, B, E, R}. (b) S = {D, I, S, C, R, E, T, E}. (c) S = {S, U, B, S, E, T, S}. 6. A school cook plans her calendar for the month of February in which there are 20 school days. Unfortunately, she only knows how to cook 4 different meals. (a) How many ways are there for her to plan her scheule of menus for the 20 school days if there are no restrictions on the number of times she cooks a particular type of meal? (b) How many ways are there for her to plan her schedule of menues if she wants to cook each meal the same number of times? 7. Twelve employees of a compay are being assigned to offices. There are four offices and each is large enough for three people. How many ways are there to assign employees to offices? 8. How many solutions are there to the equation: x1 + x2 + x3 + x4 + x5 + x6 = 25, where each xi is a non-negative integer and... (a) There are no other restructions. (b) xi ≥ 3, for i = 1, 2, 3, 4, 5, 6. (c) x1 ≥ 7 and x2 ≥ 9, (d) 3 ≤ x1 ≤ 10. (e) 3 ≤ x1 ≤ 10, 2 ≤ x2 ≤ 7, and 5 ≤ x3 . 9. How many ternary strings (digits 0,1, or 2) are there with exactly seven 0’s, five 1’s and four 2’s? 10. How many ways are there to deal hands from a standard playing deck to four players if (a) Each player gets exactly 13 cards. (b) Each player gets seven cards and the rest of the cards remain in the deck? 11. Give the characteristic equation for the following recurrence relations: (a) an = 3an−1 − an−2 + 17an−3 . (b) an = −an−2 + an−4 (c) an = 3an−7 12. Solve the following recurrence relation: • fn = fn−1 + 12fn−2 • f0 = −2 • f1 = 20 13. Solve the following recurrence relation: • fn = 3fn−1 + 4fn−2 • f0 = 4 • f1 = 1 3 14. Solve the following recurrence relation: • fn = 4fn−2 • f0 = 2 • f1 = 16 15. Solve the following recurrence relation: • fn = 4fn−1 − 4fn−2 • f0 = 3 • f1 = −10 The next two problems are just for practice. They will not be graded: 1. Solve the following recurrence relation: • fn = 3fn−1 + 4fn−2 − 12fn−3 • f0 = 4 • f1 = −5 • f2 = 11 2. Solve the following recurrence relation: • fn = 5fn−1 − 8fn−2 + 4fn−3 • f0 = 6 • f1 = 7 • f2 = 17