Illinois State University Mathematics Department

 MAT 305: Combinatorics Topics for K-8 Teachers

### Possible Solutions: Supplementary Problems Set G

Possible Solutions

Use the following generating functions and simplify each expression in (1) and (2).

1.

 (a) A + B (b) B + C (c) B + D (d) C + F

2.

 (a) AB (b) AC (c) AD (d) DE

3.

Write a generating function to determine the number of ways of choosing d drinks from a refrigerator that has 3 colas and 5 root beers. Expand your function through the 6th power.

4.

Write a generating function to determine the number of ways of choosing j jelly beans from a basket that has 3 licorice, 4 strawberry, and 2 lemon jelly beans. Expand your function through the 6th power.

5.

Write a generating function to determine the number of ways of buying p chicken parts at a grocery has 4 wings, 3 breasts, and 5 drumsticks, if the drumsticks are wrapped in a package of 2 and a package of 3 and the packages cannot be divided. Expand your function through the 6th power.

6.

Write a generating function to determine the number of ways of ordering g glasses of liquid, if 3 glasses of milk and and unlimited supply of water are available. Expand your function through the 6th power.

7.

A bakery has an unlimited supply of apple danish pastries but only three cream cheese danish pastries and only four strawberry danish pastries. The strawberry danish must be purchased two at a time.

(a) Write a generating function for the number of ways to buy r pastries under these conditions.

(b) Expand (a) through the 8th power.

(c) How many ways are there to buy 7 pastries?

The coefficient of the 7th-powered term is 12, so there are 12 ways to buy 7 pastries.

8.

Genny has a large supply of 1-cent, 2-cent, and 3-cent stamps. The stamps of each denomination are identical.

(a) Write a generating function that will determine the number of ways to arrange exactly three of the stamps in a row on an envelope so that their total value is c cents.

(b) Repeat (a) if four stamps can be used.

(c) Repeat (a) if three or four stamps can be used.

(d) Repeat (a) if any positive number of stamps can be used.

(e) Use (d) to show the number of ways to arrange 4 cents worth of stamps.

We need to expand the products, combine like terms, and look at the coefficient of the 4th-degree term. Note that we only need the first 4 terms in the infinite sum shown in (d) above.

In the expansion and after the collection of like terms, the 4th-degree term has coefficient 7. This is the number of ways to arrange 4 cents worth of stamps, using any positive number of stamps of denomination 1-, 2-, and 3-cents.

By enumeration, we can identify the particular arrangements:

 1 1 1 1 1 1 2 1 2 1 2 1 1 3 1 1 3 2 2

9.

Write a generating function to determine the number of ways to make change for a \$1 bill, given an adequate supply of nickels, dimes, quarters, and half-dollars. Do not expand your function, but do explain how you would use it to determine a solution.

After expanding this product and collecting like terms, the coefficient of the 100-degree term will be the number of ways to make the change.

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