Pick My Pin

In this activity, the students determine the number of possible combinations of coloured pegs. It is similar in nature to the activity on page 4 of the students’ book in that question 1 is a problem involving factorials.
The students may have strategies of their own (for example, a tree diagram) that they wish to try. Having six different-coloured pegs on hand would be helpful because the students will be able to see that there are six possible choices for the first peg in the code; five for the second peg (since one peg has already been allocated to the code); four for the third peg; and three possible choices for the fourth and last peg. The students could tabulate these choices in various ways, but in the end, they would find it helpful if they recognise that factorials are the most powerful and elegant approach, that is, 6 x 5 x 4 x 3 = 360.
Question 2 presents an additional challenge because here the students need to see that not only are there six possible choices for the first peg but six for each of the remaining three pegs as well; in short, the choices are 6 x 6 x 6 x 6, or 6⁴, which is 1 296.
Game
This game uses the ideas involved in the activity above. The students may ask about and wish to investigate the number of possible combinations in a four-digit code that uses all the 10 digits from 0 to 9; it would be 104 = 10 000.

Answers to Activity

1. 360. There are six choices of colour for the first peg. After you have chosen a colour for that peg, there are five colours left to choose from, which gives 30 combinations for the first two pegs. There are four choices for the third coloured peg, so there are 120 combinations. That leaves three choices for the last peg. 6 x 5 x 4 x 3 = 360
2. a. 1 296
b. Because 6 x 6 x 6 x 6 = 64, which is 1 296, and 1 296 is a lot more than the 360 you can get using different colours only.

Game
A game of logic and reasoning. Explanations for less than 100 guesses will vary but should mention reasoning strategies such as elimination (NNNN means none of those digits are in), comparison, changing all the digits, changing one digit, and changing the order of digits. Getting feedback from the person with the PIN is also crucial!