Question

Use a generating function to find the number of ways to collect $15 from 20 distinct people if each of the first 19 people can give a dollar or nothing, and the 20th person can give either nothing, $1, or $5.

Answer 1

To find the number of ways to collect $15 from 20 distinct people, we can use a generating function. The options for the 20th person include giving nothing, $1, or $5. By representing each scenario with a generating function, we can find the total number of ways by adding up the coefficients of the terms representing $15.

Step-by-step explanation:

To find the number of ways to collect $15 from 20 distinct people, we can use a generating function. Let's consider the possible options for the 20th person:

• If the 20th person gives nothing, then we are left with finding the number of ways to collect $15 from the first 19 people, where each person can give a dollar or nothing. This can be represented by the generating function: (1+x)^19.
• If the 20th person gives $1, then we need to find the number of ways to collect $14 from the first 19 people. This can be represented by the generating function: (1+x)^19 * (1+x^2+x^3+...)
• If the 20th person gives $5, then we need to find the number of ways to collect $10 from the first 19 people. This can be represented by the generating function: (1+x)^19 * (1+x^2+x^3+...)^5

To find the total number of ways, we can add up the coefficients of the terms representing $15 in each of the generating functions.