Ma2201/CS2022
Discrete Mathematics
D Term, 2013
Quiz 0110
Solutions
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1. (
4 pts
) How many 144 letter strings of lower case letters
L
=
{
a, b, c, . . . , z
}
have
either only vowels (
V
=
{
a, e, i, o, u
}
) in the the odd positions, start with 100 identical
letters, or read the same forwards as backwards.
Let
A
be the strings with vowels in the odd positions,
B
the strings which start with 100
identical letters, and
C
be the strings which read the same forwards and backwards.
We want to compute
A
∪
B
∪
C
and use inclusion exclusion.

A

= 5
72
·
26
72
, since there are 72 positions with vowels and 72 having any letter.

B

= 26
45
, since the choice of the first letter forces the next 99, leaving 44 to be
independently chosen.

C

= 26
72
, only the first 72 can be independently chosen.

A
∩
B

= 5
23
26
22
, since the first 100 letters must all be identical vowels, and the last
44 letters are half vowel, half normal.

A
∩
C

= 5
72
, if it read the same forwards and backwards, there must be vowels in all
positions, and the first 72 determine the rest.
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 Fall '10
 PeterChristopher
 Math

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