Math 106 – Combinatorics – Quiz #5 Review Sheet Name __________________________________________________ 1. (a) (b) Six refrigerators in a west coast warehouse and 12 refrigerators in an east coast warehouse must be distributed among 8 outlets. How many different ways can the west coast refrigerators be distributed among the outlets? x1 + x2 + … + x7 + x8 = 6 non-negative integers 13! ——– = 1716 7! 6! How many different ways can the east coast refrigerators be distributed among the outlets? x1 + x2 + … + x7 + x8 = 12 non-negative integers 19! ——– = 50,388 7! 12!

(c) (d) How many different ways can the west coast refrigerators be distributed among the outlets, if one outlet has been specified to receive no more than two refrigerators? x1 + x2 + … + x7 + x8 = 6 x8  2 Use GOOD = ALL – BAD x1 + x2 + … + x7 + x8 = 6 3  x8 1716 – 120 = 1596 10! ——– = 120 7! 3! How many different ways can the west coast refrigerators be distributed among the outlets, if two outlets have been specified to each receive no more than two refrigerators? x1 + x2 + … + x7 + x8 = 6 x7  2 AND x8  2 Use GOOD = ALL – BAD x1 + x2 + … + x7 + x8 = 6 3  x7 OR 3  x8 1716 – 239 = 1477 10! (2)——– – 1 = (2)(120) – 1 = 239 7! 3!

1.-continued (e) (f) How many different ways can the west coast refrigerators be distributed among the outlets, if three outlets have been specified to each receive no more than two refrigerators? x1 + x2 + … + x7 + x8 = 6 x6  2 AND x7  2 AND x8  2 Use GOOD = ALL – BAD x1 + x2 + … + x7 + x8 = 6 3  x6 OR 3  x7 OR 3  x8 1716 – 357 = 1359 10! (3)——– – (3)(1) = (3)(120) – 3 = 357 7! 3! How many different ways can the west coast refrigerators be distributed among the outlets, if two outlets have been specified to each receive no more than three refrigerators? x1 + x2 + … + x7 + x8 = 6 x7  3 AND x8  3 Use GOOD = ALL – BAD x1 + x2 + … + x7 + x8 = 6 4  x7 OR 4  x8 1716 – 72 = 1644 9! (2)——– = (2)(36) = 72 7! 2!

(g) (h) (i) How many different ways can the east coast refrigerators be distributed among the outlets, if one outlet has been specified to receive no more than two refrigerators, a second outlet has been specified to receive no more than three refrigerators, and a third outlet has been specified to receive no more than four refrigerators? Use GOOD = ALL – BAD x1 + x2 + … + x7 + x8 = 12 x6  2 AND x7  3 AND x8  4 x1 + x2 + … + x7 + x8 = 12 3  x6 OR 4  x7 OR 5  x8 50,388 – 20,066 = 30,322 16! 15! 14! 12! 11! 10! 7! —— + —— + —— – —— – —— – —— + —— = 7! 9! 7! 8! 7! 7! 7! 5! 7! 4! 7! 3! 7! 0! 11,440 + 6435 + 3432 – 792 – 330 – 120 + 1 = 20,066 How many different ways can the east coast refrigerators be distributed among the outlets, if all outlets can receive no more than one refrigerator? 0 (zero), since this is impossible How many different ways can the west coast refrigerators be distributed among the outlets, if all outlets can receive no more than one refrigerator?

x1 + x2 + … + x7 + x8 = 6 x1  1 AND x2  1 AND … AND x8  1 1716 – 1688 = 28 Use GOOD = ALL – BAD x1 + x2 + … + x7 + x8 = 6 2  x1 OR 2  x2 OR … OR 2  x8 11! 9! 7! (8)—— – (28)—— + (56)—— = (8)(330) – (28)(36) + (56)(1) = 1688 7! 4! 7! 2! 7! 0!