Problem In playing cards what is the probability to get exactly one pair (for example (1,1), (2,2)) if we draw 5 cards Solution Step one is to compute how many possibilities we have if we draw 5 cards without any restriction. The answer is simply 52 choose 5 which is given by the well known

Subsets Using Binomial Theorem How many subsets we can make from the numbers between 1 and 5 Solution The naive way to compute that is to enumerate all possibilities as follows: Groups of 1: 1, 2, 3, 4, 5 Groups of 2: 12, 13, 14, 15, 23, 24, 25, 34, 35, 45 Groups of 3:

Problem Given four binary digits all ones 1111 and two binary digits all zeros 00 we need to create a six digit binary number using these ones and zeros such that no two zeros are next to each other. How many binary numbers we can make. For example 011110 is correct while 110011 is not

Problem A group of 4 boys and 3 girls. We want to form a team of 5 players. We are required to select 2 girls and 3 boys. If 2 boys can not play together. How many possible ways we can form the team. Solution Let us break this problem into steps. The first step

Problem You are given 1 red flag, 2 green flags and 3 blue flags. You are asked to generate 6 symbol based signals using these flags. How many signals you can generate. Solution This problem can be solved using permutations counting techniques. We have 6 symbols in total but note that they are not distinct.

Problem A typical car license plate number in California starts with a decimal digit then three capital letters then three decimal digits. If repetition is not allowed and we need to generate a license plate number randomly, what is the probability that we will get a three letter sequence "ABC" in the generated license plate

Problem Consider the system where we have a chain of 4 antennas lined up next to each other. In order for the system to be functional no two defective antennas should be lined up next to each other. Given that 2 antennas are defective, what is the probability that the system is functional. Solution We

Problem In a parking lot there are (4) trucks, (3) SUVs, (2) Sedans and one motorcycle. We need to park the cars such that cars of the same type are next to each other. For example all trucks are adjacent to each other, SUVs are next to each other and so on. How many different

Triangle ants problem Three ants on the corners of an equilateral triangle. They started moving along the edges at the same speed. What is the probability that the ants will collide ? Solution Each ant could possibly be moving in one direction let us say Left or Right. We have three ants with two possible