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1. Show
that
n
n
2
!
for all
4
≥
n
by induction.
2. Show that
1
2
!
+
n
n
for all
5
n
by induction.
n
k
n
2
!
3. Use induction to show that
n
straight lines in the plane, with no two parallel and no
three going
through a common point, divide the plane into
2
/
)
2
(
2
+
+
n
n
regions.
4.
Show that the regions of the preceding exercise can be colored red and green so that
no two regions sharing a common edge have the same color.
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Unformatted text preview: 5. A 3Dseptomino is a threedimensional 2 2 2 × × cube with one of the little 1 1 1 × × corner cubes removed. Prove that a n n n 2 2 2 × × cube with any one of the 1 1 1 × × cubes can be tiled by 3Dseptominoes. 6. Prove by induction that 1 1 4 3 3 2 2 1 2 + < + + ⋅ ⋅ ⋅ + + + n n n n ....
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This note was uploaded on 08/24/2009 for the course MATH 262447221 taught by Professor Weisbart during the Spring '09 term at UCLA.
 Spring '09
 WEISBART

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