Originally Posted by

**SlipEternal** For questions like this, you should consider all possible outcomes. Here are the outcomes for the first problem:

You have three balls, each a different color.

You have at least two balls of the same color.

An event and its opposite are mutually exclusive. Since there are a total of 15 balls, the total number of choosing possibilities is the number of ways to choose 3 balls from a bucket of 15 balls. Since each outcome is distinct and constitute all possible outcomes, the sum principle says that the sum of the choosing possibilities for these two will give you the total number of choosing possibilities.

Same thing goes for question 2. Consider the opposite of what it is asking. Which is easier to count? In this case, the opposite is easier to count. How many six-character words contain no two identical characters? That is equivalent to asking how many six-character words contain six distinct characters? Subtract that from the total number of possible words (with no restrictions) to find the number of different words the problem is asking.

For question 3, try breaking the problem down into easier pieces. For any chosen group, you can line the members up and assign jobs to the first three people in line in order: leader, treasurer, and spokesman. The fourth group member would not have a job. So, the number of ways to choose the group is independent from the number of ways to choose the jobs for the group members. This allows you to use the product principle. It is the number of ways to choose the members of the group times the number of ways to give the jobs to the members.