Unformatted text preview: 68 1. ALGEBRAIC THEMES 1.9.4. Prove that
n.n n2 1/2 D n
X k.k k D0 Use this to ﬁnd a formula for Pn k D0 k2 ÂÃ
n
1/
:
k ÂÃ
n
.
k 1.9.5. Bernice lives in a city whose streets are arranged in a grid, with
streets running northsouth and eastwest. How many shortest paths are
there from her home to her business, that lies 4 blocks east and 10 blocks
north of her home? How many paths are there which avoid the pastry shop
located 3 blocks east and 6 blocks north of her home?
1.9.6. Bernice travels from home to a restaurant located n blocks east and n
blocks north. On the way, she must pass through one of n C 1 intersections
located n blocks from home: The intersections have coordinates .0; n/,
.1; n 1/, . . . , .n; 0/. Show that the number of paths to the restaurant
Â Ã2
n
that pass through the intersection with coordinates .k; n k / is
.
k
Conclude that
n
X ÂnÃ2 Â2nÃ
D
:
k
n
k D0 1.9.7. How many natural numbers Ä 1000 are there that are divisible by
7? How many are divisible by both 7 and 6? How many are not divisible
by any of 7, 6, 5?
1.9.8. A party is attended by 10 married couples (one man and one woman
per couple).
(a) In how many ways can the men and women form pairs for a
dance so that no man dances with his wife?
(b) In how many ways the men and women form pairs for a dance so
that exactly three married couples dance together?
(c) In how many ways can the 20 people sit around a circular table,
with men and women sitting in alternate seats, so that no man
sits opposite his wife? Two seating arrangements are regarded
as being the same if they differ only by a rotation of the guests
around the table.
1.9.9. Show that Fermat’s little theorem is a special case of Euler’s theorem.
The following three exercises outline a proof of Euler’s theorem.
1.9.10. Let p be a prime number. Show that for all integers k and for all
nonnegative integers s ,
s .1 C kp/p Á 1 .mod p s C1 /: ...
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 Fall '08
 EVERAGE
 Algebra, Natural number, Prime number, Bernice

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