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FINAL EXAMINATION
FALL TERM 2002
QUESTION 1.
(a) [3] Prove or disprove: in a nonempty universe of dis
course,
(
∀
x
∃
y P
(
x,y
))
⇒
(
∃
y
∀
x P
(
x,y
))
.
(b) [5] Let
a
n
be deﬁned by
a
0
= 1 and, for
n
≥
1,
a
n
=
√
a
n

1
+ 5. Prove
∀
n
≥
0,
a
n
≤
(1 +
√
21)
/
2.
QUESTION 2:
[5] Find all solutions to 15
x
+ 65
y
= 150.
QUESTION 3:
A standard deck of cards has 52 cards, 13 in each of four
suits.
(a) [5] What is wrong with the following argument that “shows” there
are 4
±
39
13
²
ways to select 13 cards from the 52 so that at least one suit has no
cards among the 13 selected.
“For any speciﬁc one of the suits, there are 39 cards not in that suit. Thus,
there are
±
39
13
²
ways of selecting 13 cards so that none are in the speciﬁc suit.
Since there are 4 ways of selecting the speciﬁc suit, there are 4
±
39
13
²
ways
of selecting 13 cards so that at most three suits appear among the selected
cards.”
(b) [5] Give a correct count for the number of ways of selecting 13 cards
from the standard deck so that at most three suits appear among the selected
cards.
QUESTION 4:
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 Spring '11
 Nayak

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