Solutions to Homework Six
CS 191 :
Discrete Structures I
Apr. 17
,
Winter 2007
Due Date: Apr. 26, 2007, Thursday, before the class meeting.
Section 6.6, p265
In Exercise 13, determine the number of strings that can be formed by ordering the
letters given.
Exercise
(2)
SCHOOL
Answer
:
There are
1
S,
1 C,
1 H,
2 Os,
1 L
Therefore, the answer is
!
2
!
6
=360.
Exercise
(5)
How many strings can be formed by ordering the letters
SALESPERSONS
if no two S’s
are consecutive?
Answer
:
We consider a twostep procedure.
(1)
We form strings in which no S occurs. This can be done in (8!/2!) ways.
(2)
We then place the four S’s in the seven inbetween positions and two ending
positions. This can be done in C(9, 4) ways.
Therefore, the answer is
C(9, 4)
×
8!/2! =
!
4
6
7
8
9
×
×
×
×
8!/2! = 2,540,160.
Exercises 1521 refer to piles of identical red, blue, and green balls where each pile
contains at least 10 balls.
1
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Exercise
(19)
In how many ways can 10 balls be selected if exactly one red ball and at least one blue
ball must be selected?
Answer
:
This problem is equivalent to how many ways we can select 9 balls from the blue balls
and green balls with a condition that at least one blue ball must be selected. This question
is also equivalent to how many ways we can select 8 balls from the blue balls and green
balls. Therefore, the answer is
C(8+21, 8) = C(9, 1) = 9.
In Exercises 2229, find the number of integer solutions of
x
1
+
x
2
+
x
3
= 15
subject to the conditions given.
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 Winter '06
 Shen
 Combinatorics, Finite set

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