“Typical IB Problems”
1 10
)
2x
Find the coefficient of x8 in this expression.
10 10
 
1
1
(3x 2  )10    (3x 2 ) K ( )10k
2x
2x
K 0  k 
(3 x 2 
10  K 2 K (1)10 K
  3 x
210 K x10 K
K
10  3 K (1)10 K
  
 x 3 K 10
10 K
K 2
10  36
   4
 6 2
210  729

 9568.125
16
3K  10  8
K 6
1
(2 x  ) 8
x
Find the constant term (before x0)
8
8
1
1
(2 x  ) 8    (2 x) K ( ) 8 K
x
x
K 0  K 
8
  2 K (1) 8 K x 2 K 8
K
8
  2 4
 4
2K  8  0
K 4
In terms of a what is the coefficient to x2?
(1 2x) 5 (1 ax) 6
5
5
6
6
(1  (2x)1   (2x) 2  ...)(1  ax   ( ax) 2  ...)
1
2
1
2
6 2 2 5 6
5 2
 a x   2 x ax   4 x  ....
2
1 1
2
15a 2  60a  40

Pretend we live in N.Y.C.. You need to go 8 blocks east and 15 blocks north. There is a
street every block. How many ways can you get from home to work if you have to go
only north and east?
23 23
Answer:   or  
15  8 
The way to see this is consider giving directions. We could do this by a string consisting
of 8 Es and 15 Ns. For example, EEEEEENNNNNEENNNNNNNNNN would
correspond to going 6 blocks east, follows by 5 north, then 2 east, and 10 north. Every
distinct
possible string corresponds with a different path. Thus, the number of paths is the
23!  23 
same as the total number of permutations of 8 Es and 15 Ns, which is
 .
8!15!  15 
If you must shoot the bottom-most target, how many ways can you shoot all of the
targets?
Red
Green
9!
4!2!3!
RGGBRBRBR
Tells you exactly which one to hit
Answer:

Total number of objects/frequency of each object
Blue