Unizor - Combinatorics - Advanced Problems 2.3.
Channel: Zor Shekhtman on UNIZOR Education
Date: 2014-06-20

Problem 1
How many six digit natural decimal numbers contain three odd and three even digits?
Answer: 281,250

Problem 2
Find natural numbers X and Y if
C(X+1,Y+1) : C(X+1,Y) : C(X+1,Y−1) = 5:5:3
Answer: X=6, Y=3

Problem 3
Two basketball teams, Green Shirts and Blue Shirts, are playing against each other on neutral territory. There are K fans, neutral in the beginning, watching the game.
Some M of them definitely will not take the Green Shirts' side, but might stay neutral or take the Blue Shirts' side.
Some other N of them definitely will not take the Blue Shirts' side, but might stay neutral or take the Green Shirts' side.
Remaining K−M−N fans are free to take either side or stay neutral.
How many different distributions of fans among the teams are possible?
Answer: 2^(M+N)·3^(K−M−N)

Problem 4
Consider a sphere and a set of planes going through its center. Each plane, by itself, divides the volume of a sphere in two parts. Assume that our set of planes contains N such planes and that no three planes go through the same diameter of a sphere.
In how many pieces the volume of this sphere is divided by these N planes?
Hint: Review Advanced Combinatorics Problems 1.5.
Answer: 2+N·(N−1)