Math 15B (Bender)
Midterm Exam
27 October 2000
1. (36 pts) A
ﬁve digit number
is a sequence of ﬁve digits, the ﬁrst of which is
NOT
zero. Thus, 12345 and 10101 are valid but 01234 and 1234 are
NOT
valid.
(a) How many ﬁve digit numbers are there?
Ans:
One way is to note there are 9 choices for the ﬁrst digit and 10 for each of the
remaining giving 9
×
10
4
. You could also note that it is simply all numbers less
than 10
5
and at least 10
4
giving 10
5

10
4
=9
×
10
4
.
(b) How many ﬁve digit numbers have all digits diﬀerent? (as in 12345 but not 10101)
Ans:
There are 9 choices for the ﬁrst digit, 9 for the second (anything but the ﬁrst), 8
for the third (anything but the ﬁrst two), and so on, giving 9
×
9
×
8
×
7
×
6.
(c) How many ﬁve digit numbers have no digit appearing just once? (So 11111 and
10010 are okay, but 10111 and 12312 are not.)
Ans:
Either one digit appears 5 times OR two digits appear, one twice and one three
times. There are 9 possibilities for the ﬁrst case. For the second case, choose
the two positions for the pair of digits in
(
5
2
)
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.
 Fall '01
 Bender
 Math, Probability theory, Numerical digit, ﬁrst digit

Click to edit the document details