An apartment block has 8 tenants, each with
a separate mail slot. Eleven different pieces
of junk mail are delivered.
(a) (4 points) How many ways can the
mail be delivered if three tenants get
two pieces of junk mail and five get one
piece?
Suppose the mail slots are numbered 1,2,3,4,5,6,7,8
Pick the 3 mail slots to get 2 pieces in C(8,3) ways.
Pick the 2 pieces of junk mail to put in the lowest numbered
mail slot that gets 2 pieces in C(11,2) ways.
Pick the 2 pieces of junk mail to put in the next lowest numbered
mail slot that gets 2 pieces in C(9,2) ways.
Pick the 2 pieces of junk mail to put in the highest numbered
mail slot that gets 2 pieces in C(7,2) ways.
The remaining 5 pieces of mail can be delivered to the remaining
5 boxes in P(5,5) or 5! or 120 ways.
Answer: C(8,3)×C(11,2)×C(9,2)×C(7,2)×5! = 56×55×36×21×120 = 279417600
(b) (5 points) In how many ways can the
mail be delivered if the only restriction
is that every tenant gets at least one
piece of mail?
We can distribute the 11 pieces of junk mail these ways
1. 4 in one box, 1 each in the other 7 boxes
pick the box to get four in 8 ways.
Pick the pieces of junk mail to go in it C(11,4) ways.
Pick the way to place the 7 remaining pieces of junk mail 7! ways.
That's 8×C(11,4)×7! = 8×330×5040 = 13305600
2. 3 in one box, 2 in one box and 1 each in the other 6 boxes
pick the box to get 3 in 8 ways.
pick the box to get 2 in 7 ways.
pick the pieces of junk mail to go in the box that is to get 3 in
C(11,3) ways.
pick the pieces of junk mail to go in the box that is to get 2 in
C(8,2) ways.
Pick the way to place the 6 remaining pieces of junk mail 6! ways.
That's 8×7×C(11,3)×C(8,2)×6! = 8×7×165×28×720 = 186278400
3. 2 in one box, 2 in another box and 2 in a third box and 1 each in the
other 5 boxes
pick the three boxes to get 2 in C(8,3) ways.
pick the 2 pieces of mail to go in the lowest numbered box of those
three that get two in C(11,2) ways
pick the 2 pieces of mail to go in the next higher numbered box of those
three that get two in C(9,2) ways
pick the 2 pieces of mail to go in the highest numbered box of those
three that get two in C(7,2) ways
Pick the way to place the 5 remaining pieces of junk mail 5! ways.
That's C(8,3)×C(11,2)×C(9,2)×C(7,2)×5! = 56×55×36×21×120 = 249417600
Answer: Total = 13305600+186278400+249417600 = 479001600
Edwin
