AMS 301.2
(Spring 2011)
Homework Set # 8 – Solution Notes
# 10, 5.3: (a). We first give each person a pear, and now distribute 5 identical apples and 3
identical pears to 3 distinct people:
parenleftbigg
5 + 3

1
5
parenrightbiggparenleftbigg
3 + 3

1
3
parenrightbigg
(b). The apples can be distributed 3
5
ways, as each apple has 3 choices. The pears can be split
either 1,1,4 (3 ways to choose who gets 4 pears) or 1,2,3 (3! ways to pick who gets 1 or 2 or 3
pears) and 2,2,2. Once we have the split, we can use Theorem 1. The final answer is:
3
5
(3
P
(6;1
,
1
,
4) + 3!
P
(6;1
,
2
,
3) +
P
(6;2
,
2
,
2))
# 12, 5.3: Here we break into cases depending on
k
∈ {
0
,
1
,
2
,
3
}
, the number of balls we select
out of the pink, lavender, and tan balls. This can be done in
(
3
k
)
ways. Then we must select the
remaining 8

k
balls out of 3 types (red, white or blue), which can be done in
(
8

k
+3

1
8

k
)
ways
(there are 8

k
“x’s” and 3

1 = 2 “slashes”).
k
can be any number between 0 and 3, so we sum
over those possibilities to get the final answer:
3
summationdisplay
k
=0
parenleftbigg
3
k
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '11
 Estie
 Pick, BMW Sports Activity Series, BMW X5

Click to edit the document details