Probability

Generating Functions

Generating Functions: Level 4 Challenges

         

Cody has 4 types of onions:

  • The number of purple\color{#69047E}\text{purple} onions can be any non-negative integer.
  • The number of green\color{#20A900}\text{green} onions is a multiple of 2.
  • The number of red\color{#D61F06}\text{red} onions is a multiple of 3.
  • The number of blue\color{#3D99F6}\text{blue} onions is a multiple of 5.

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

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 NN possible ways the bunny can be on the number 10 after 10 minutes. Find the last three digits of NN.

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}\{0000022222\} and {2020202020}\{2020202020\} are considered distinct.

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

Consider the recurrence relation an=2an1+3an2+3n a_n = 2a_{n-1} + 3 a_{n-2} + 3^n for n2n \geq 2 with initial conditions a0=1,a1=1.a_0 = -1, a_1 = 1.

Given that a100a_{100} is in the form of

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

where x,y,w,zx,y,w,z are prime numbers and vv as a perfect square, what is the value of w+v+x+y+zw+v+x+y+z?

Find the general expression for cnc_n, where

cn3cn1=25n. c_n - 3 c_{n-1} = 2 \cdot 5^n .

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