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Generating Functions

This combinatorial tool is used for solving recurrence relations and other difficult problems.

Challenge Quizzes

Level 4

Cody has 4 types of onions.

\(\bullet\) The number of \(\color{Purple}{Purple}\) onions can be any non-negative integer.

\(\bullet\) The number of \(\color{Green}{Green}\) onions is a multiple of 2.

\(\bullet\) The number of \(\color{Red}{Red}\) onions is a multiple of 3.

\(\bullet\) The number of \(\color{Blue}{Blue}\) onions is a multiple of 5.

If Cody has 23 onions, how many different distributions of colors can there be?

Details and Assumptions: Cody can't have a negative number of onions, but he may have 0 onions of a certain type(s).

A one year old bunny is sitting on the number 0 in the number line. His father, Bugs Bunny, is waiting for him on the number 10.

The bunny has to reach his father. At each minute, he can:

  • Move one step forward;
  • Move two steps forward;
  • Stay still;
  • Move one step backward;
  • Move two steps backward.

There are \(N\) possible ways the bunny can be on the number 10 after 10 minutes. Find the last three digits of \(N\).

Details and assumptions

  • The bunny is allowed to hop beyond 10, and then come back.

  • There are no restrictions on the bunny entering the negative numbers. The bunny can go as much backwards as he wants.

  • Order does matter. For example, the steps \(\{0000022222\}\) and \(\{2020202020\}\) are considered distinct.

How many polynomials \( P(x) \) are there such that the coefficients of \( P(x) \) are integers from 0 to 24 (inclusive) and \( P(5) = 1200 \)?

Consider the recurrence relation \( a_n = 2a_{n-1} + 3 a_{n-2} + 3^n \) for \(n \geq 2 \) with initial conditions \(a_0 = -1, a_1 = 1. \)

Given that \(a_{100}\) is in the form of

\[ \LARGE \frac {x\cdot y^z - w}{v} \]

where \(x,y,w,z\) are prime numbers and \(v\) as a perfect square, what is the value of \(w+v+x+y+z\)?

Find the general expression for \(c_n\), where

\[ c_n - 3 c_{n-1} = 2 \cdot 5^n . \]

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