Math 106 – Combinatorics – Quiz #4 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) (e) How many different ways can the west coast refrigerators be distributed among the outlets, if two outlets have been specified to each receive no refrigerators? x1 + x2 + … + x5 + x6 = 6 non-negative integers 11! ——– = 462 5! 6! How many different ways can the east coast refrigerators be distributed among the outlets, if three outlets have been specified to each receive exactly two refrigerators? x1 + x2 + … + x7 + x8 = 12 x6 = 2 & x7 = 2 & x8 = 2 x1 + x2 + x3 + x4 + x5 = 6 non-negative integers 10! ——– = 210 4! 6! How many different ways can the west coast refrigerators be distributed among the outlets, if three outlets have been specified to each receive exactly two refrigerators? x1 + x2 + … + x7 + x8 = 6 x6 = 2 & x7 = 2 & x8 = 2 x1 + x2 + x3 + x4 + x5 = 0 non-negative integers 4! ——– = 1 4! 0!

1.-continued (f) (g) (h) How many different ways can the west coast refrigerators be distributed among the outlets, if one outlet has been specified to receive at least four refrigerators? x1 + x2 + … + x7 + x8 = 6 x8  4 x1 + x2 + … + x7 + x8 = 2 non-negative integers 9! ——– = 36 7! 2! How many different ways can the east coast refrigerators be distributed among the outlets, if two outlets have been specified to each receive at least three refrigerators? x1 + x2 + … + x7 + x8 = 12 x7  3 & x8  3 x1 + x2 + … + x7 + x8 = 6 non-negative integers 13! ——– = 1716 7! 6! How many different ways can the west coast refrigerators be distributed among the outlets, if one outlet has been specified to receive at most two refrigerators? x1 + x2 + … + x7 + x8 = 6 x8  2 x1 + x2 + … + x7 + x8 = 6 x8  3

x1 + x2 + … + x7 + x8 = 3 non-negative integers 10! ——– = 120 7! 3! 1716 – 120 = 1596 (i) (j) (k) How many different ways can the east coast refrigerators be distributed among the outlets, if one outlet has been specified to receive at least two refrigerators but not more than five refrigerators? x1 + x2 + … + x7 + x8 = 12 2  x8  5 x1 + x2 + … + x7 + x8 = 12 2  x8 x1 + x2 + … + x7 + x8 = 12 2  x8 & 6  x8 17! ——– = 19,448 7! 10! 13! ——– = 1716 7! 6! 19,448 – 1716 = 17,732 How many different ways can the east coast refrigerators be distributed among the outlets, if one outlet has been specified to receive at least two refrigerators, and another outlet has been specified to receive not more than five refrigerators? x1 + x2 + … + x7 + x8 = 12 2  x7 & x8  5 x1 + x2 + … + x7 + x8 = 12 2  x7 x1 + x2 + … + x7 + x8 = 12 2  x7 & 6  x8 17! ——– = 19,448 7! 10! 11! ——– = 330 7! 4! 19,448 – 330 = 19,118 How many different ways can the west coast refrigerators be distributed among the outlets, if each one of the outlets must receive at least one refrigerator? x1 + x2 + … + x7 + x8 = 6 positive integers zero (0), not possible