3D Tic-Tac-Toe

In how many ways can you get three in a row in three-dimensional tic-tac-toe (noughts and crosses)?

Lazy student

A student guesses randomly on a 50-question True/False test. Find the probability that the number of correct guesses is divisible by 4.

Random subsets

Two random subsets A and B are selected from an n-element set. What is the probability that A is a subset of B?

Numbers less than one trillion containing '99'

How many numbers less than one trillion (10^12) have two or more consecutive nines in their digits?

(x+y+z)^100

How many terms does (x+y+z)^100 have when expanded? (After combining like terms.)

Writing 20 as a sum of odd numbers

There are three ways to write 10 as the sum of four odd positive integers, assuming that order does not matter. They are 1+1+1+7, 1+1+3+5, and 1+3+3+3. In how many ways can 20 be written as the sum of eight odd positive integers?

Source: The Moscow Puzzles, by Boris Kordemsky.

Houses on a circular road

Nine houses are built on a circular road. Each house is to be painted either red, white, or blue. Adjacent houses must have different colors. How many color combinations are possible?