Illinois State University Mathematics Department
MAT 305: Combinatorics Topics for K-8
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Possible Solutions: Supplementary Problems Set G |
return to Set G problems |
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Use the following generating functions and simplify each expression in (1) and (2). ![]() |
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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. ![]() |
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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. ![]() |
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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. ![]() |
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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. ![]() |
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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. |
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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:
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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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