Problem Set 3
Math-CSCI 2011 - Normandale Community College
2022-06-13
Problem 1 In how many ways can a photographer at a wedding arrange 6 people in a row from a
group of 10 people, where the bride and the groom are among these 10 people, if
1. the bride must be in the picture?
2. both the bride and groom must be in the picture?
3. exactly one of the bride and the groom is in the picture?
.
2.
3.
si-aazo
(Y).6!=
(i).
6:
(8) .:
y4.4.s1=
souo
=
?*si=40z
=
1
Problem 2 How many different functions are there from a set with 10 elements to sets with the
following numbers of elements?
• 2 210 1024
• 3 - ,10 59049
• 4
418
1048376
• 5
9165625
=
=
=
=
=
510
=
=
2
Problem 3 Let (xi , yi , zi ), i = 1, 2, 3, 4, 5, 6, 7, 8, 9, be a set of nine distinct points with integer coordinates in xyz-space. Show that the midpoint of at least one pair of these points has integer
coordinates.
midpointbetween
midpointwill
Lodd,
or
be int
odd, oddly
+
seven, even,
every
=
2 8
since
possibilities
there
are
(a, b, and St, a-**,
a
when
points
both
Codd, oddlyodoy-
even, even,
for
order
every (in
arranying
triples,
link,
one
odal
of
are
it
or
2
both odd
into
it, into
andeven
their
even
**
bury
midpoints
in
will
3
coordinat
be
all
inte
Problem 4 Show that if f is a function from S to T , where S and T are finite sets with |S| > |T |,
then there are elements s1 and s2 in S such that f (s1 ) = f (s2 ), or in other words, f is not one-to-one.
since
more
one
the
function
elements
in
send
$,
4
elements
from
isnot
it
one to
of
element
with
telements
4
5 to 4
one
of
is
5,
and
there
so
are
there
is
attenst
f(x) f(z)
=
Problem 5 In how many different orders can five runners finish a race if no ties are allowed?
18)
15: =
=
iz
5
Problem 6 How many ways are there for 10 women and six men to stand in a line so that no two
men stand next to each other?
Hint: First position the women and then consider possible positions for the men.
(*!=
1628800
Cr=
sis:*!=
172,84w
332640x3628800 1,267,084,072,000
=
6
Problem 7 Suppose that a department contains 10 men and 15 women. How many ways are there
to form a committee with six members if it must have more women than men?
1
I men
women
i!" snas
(i).)?
S women
=
I
man
(s)(Y)=sit!":=20020
6
women
(3)
Omen
4
=
s00s
=
61423 36838 5005
+
+
96460
=
7
Problem 8 Show that there are
!m+n"
n
paths between point (0, 0) and (m, n) in the xy-plane.
8
Problem 9 There are 51 houses on a street. Each house has an address between 1000 and 1099,
inclusive. Show that at least two houses have addresses that are consecutive integers.
100
numbers
+Y 1.96032
=
51
horses
since
theme
is
an
average
ofless
then
2
hors per
he
9
heat
1,
to
each
I most
other
Problem 10 Show that there are
satisfying the equation
!n≠1"
r≠1
distinct positive integer-values sequences (x1 , x2 , · · · , xr )
x1 + x2 + · · · + xr = n
10
xi > 0, i = 1, · · · , r
Problem 11 If 8 identical whiteboards are to be divided among 4 schools, how many divisions are
possible? How many if each school must receive at least one whiteboard?
+,
+2 12+y =
+
1.58 ()
=
2.(3) i7
=
=
=
163
3
=
11
Problem 12 How many 5-card poker hands are there?
(3) =2998950
=
12
Problem 13 A student is to answer 7 out of 10 questions in an exam.
1. How many choices does she have?
2. How many if she must answer at least 3 out of the first 5 questions?
"...
350
=
13
Problem 14 There are 13 students in a class. In how many ways can we selecct a committee of any
size and a chairperson for it?
Hint: Let k be the committee size, then k = 1, 2, · · · , 13 with one member being the chairperson.
Find number of such committes for each k then add them.
,k(i)
k
I
0192
=
1
=
did
the
computer
calculating
CO2
in a
on
lot
·
14
Problem 15 How many bit strings of length 10 have
1. Exactly three 0’s?
2. More 0’s than 1’s?
3. At leat seven 1’s?
4. At least three 1’s?
1.
2.
b.
(210,2)
=
120
=
((10,41 C(10,2) CC10,2
+
+
+
2(10,4) ((10,2) ((10,y
((10,0)
CC10,1
10
=
+
is:=
so
+
+
+
4.2(10,3 k(10,4 2(10,4
+
+
sid,0)
+
k6
=
(((0,) (((0,0) (((0,a e(10,10) 938
+
+
+
+
=
15
Problem 16 How many bit strings contain exactly eight 0’s and 10 ones if every 0 must be immediately
followed by a one?
0181010181 01
(19,2)
1
=
01 011
1
36 a 4)
=
+
=
16