ch6 - Section 6.1 Generating Function models Def: Suppose a...

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Unformatted text preview: Section 6.1 Generating Function models Def: Suppose a r is the number of ways to select r objects in some procedure. Then g ( x ) is a generating function for a r if g ( x ) = a + a 1 x + a 2 x 2 + a r x r + Example: (1 + x ) n = 1 + parenleftBigg n 1 parenrightBigg x + parenleftBigg n 2 parenrightBigg x 2 + parenleftBigg n r parenrightBigg x r + parenleftBigg n n parenrightBigg x n Example: (1 + x + x 2 ) 4 Example 1: Find the generating function for a r the number of ways to select r balls from 3 green, 3 white, 3 blue, and 3 gold. Example 2: Find the generating function for a r the number of ways to select r doughnuts from 5 chocolate, 5 strawberry, 3 lemon, 3 cherry? Reapeat with at least 1 per type. Example 3: Find the generating function for a r the number of ways to select r objects from 3 types with at most 4 per type. What if there is no limit on the number from each type?...
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ch6 - Section 6.1 Generating Function models Def: Suppose a...

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