In this activity, students explore the combinations and permutations of a Pick Three lottery game. In each round, students choose a three-digit number and the teacher generates a random three-digit -- winning -- number. In this game, the order of the numbers matters. Matching all three digits in the correct order wins 250 points. Matching all three digits in any order wins 50 points. After 20 rounds, the students calculate the probability of the winning combinations and permutations. They also calculate how many tickets they would need to buy to guarantee a win.
Creating a combinations tower provides students with a visual representation of combinations and permutations. The tower is built using three-by-three grids with each of the nine squares colored either blue or green. The goal is to create all possible permutations of the color patterns and group them by their combinations. For example, the second column would contain all patterns with one green square and eight blue squares. The challenge of this activity is to avoid duplicating the permutations.
In this activity, five students are given one of the following cards: ace, two or three of hearts, or four or five of spades. The students then stand at the front of the room and rearrange themselves to answer questions about combinations and permutations. For example, how many pairs can be made from all five cards, how many arrangements can be made from the three heart cards, or how many arrangements can be made from the three heart cards if the order of the cards matters? Students will use the visual model as a starting point for calculating the combinations and permutations.
Building a sandwich involves creating combinations using the sandwich ingredients. In this activity, students choose from four fillings -- such as roast beef, cheese, pickles and mayonnaise -- to build a sandwich with only three fillings. Using this visual model, students calculate the possible combinations. In the first activity, the order does not matter, but students can repeat the activity assuming that order is important. They will then calculate the permutations.