Math 106 – Combinatorics – Quiz #4 Review Sheet
Name __________________________________________________
1.
Six refrigerators in a west coast warehouse and 12 refrigerators in an east coast warehouse
must be distributed among 8 outlets.
(a) How many different ways can the west coast refrigerators be distributed among the
outlets?
x1 + x2 + … + x7 + x8 = 6
13!
——– = 1716
non-negative integers
7! 6!
(b) How many different ways can the east coast refrigerators be distributed among the
outlets?
x1 + x2 + … + x7 + x8 = 12
19!
——– = 50,388
non-negative integers
7! 12!
(c) 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!
(d) 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!
(e) 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
x1 + x2 + x3 + x4 + x5 = 0
x6 = 2 & x7 = 2 & x8 = 2
non-negative integers
4!
——– = 1
4! 0!
1.-continued
(f) 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
x1 + x2 + … + x7 + x8 = 2
x8  4
non-negative integers
9!
——– = 36
7! 2!
(g) 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
x1 + x2 + … + x7 + x8 = 6
x 7  3 & x8  3
non-negative integers
13!
——– = 1716
7! 6!
(h) 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
x1 + x2 + … + x7 + x8 = 6
x8  2
x8  3
x1 + x 2 + … + x7 + x8 = 3
non-negative integers
10!
——– = 120
7! 3!
1716 – 120 = 1596
(i) 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
x1 + x2 + … + x7 + x8 = 12
2  x8  5
2  x8
x1 + x2 + … + x7 + x8 = 12
x1 + x2 + … + x7 + x8 = 12
x1 + x2 + … + x7 + x8 = 12
2  x7 & x8  5
2  x7
2  x7 & 6  x8
x1 + x2 + … + x7 + x8 = 12
2  x8 & 6  x8 13!
——– = 1716
7! 6!
19,448 – 1716 = 17,732
(j) 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?
17!
——– = 19,448
7! 10!
17!
——– = 19,448
7! 10!
11!
——– = 330
7! 4!
19,448 – 330 = 19,118
(k) 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