Combinatorics Through Guided Discovery, 2004a

183
from the row above. Even in this situation, there are certain slight additional
assumptions you need to make, so this hint leaves you a lot of work to do. (It
is reasonable to expect problems because of that exceptional case.) However,
it should lead you in a useful direction.
183. Substitute something for A, P and B in your formula from Problem 181.
184. For example, to get the cost of the fruit selection APB you would want to get
x 20 x 25 x 30 = x 75 .
186. Consider the example with n =2. Then we have two variables, x 1 and x 2 .
Forgetting about x 2 , what sum says we either take x 1 or we don’t? Forgetting
about x 1 , what sum says we either take x 2 or we don’t? Now what product
says we either take x 1 or we don’t and we either take x 2 or we don’t?
188. For the last two questions, try multiplying out something simpler first, say
(a 0 +a 1 x +a 2 x 2 )(b 0 +b 1 x +b 2 x 2 ) . If this problem seems difficult the part that
seems to cause students the most problems is converting the expression they
get for a product like this into summation notation. If you are having this
kind of problem, expand the product (a 0 + a 1 x + a 2 x 2 )(b 0 + b 1 x + b 2 x 2 ) and
then figure out what the coefficient of x 2 is. Try to write that in summation
notation.
189. Write down the formulas for the coefficients of x 0 , x 1 , x 2 and x 3 in
(
∑ n
)
a i x i
m∑
b j x j .
i=0
j=0
190. How is this problem different from Problem 189? Is this an important difference
from the point of view of the coefficient of x k ?
191. If this problem appears difficult, the most likely reason is because the definitions
are all new and symbolic. Focus on what it means for ∑ ∞
k=0 c kx k to
be the generating function for ordered pairs of total value k. In particular,
how do we get an ordered pair with total value k? What do we need to know
about the values of the components of the ordered pair?
192.b. You might try applying the product principle for generating functions to an
appropriate power of the generating function you got in the first part of this
problem.
Additional Hint: In Problem 125 you found a formula for the number of
k-element multisets chosen from an n-element set. Suppose you use this
formula for a k in ∑ ∞
k=0 a kx k . What do you get the generating function for?
195. While you could use calculus techniques, there is a much simpler approach.
Note that 1+x =1− (−x).