Keegan needs to choose a code for her safe. She is allowed to pick a code between 3 and 8 digits long (a "digit" is any of the numbers 0 through 9). Keegan wants to keep the code as short as possible while maintaining good security. What is the least number of digits Keegan can pick for her code so that a potential thief has no more than a 0.0001% chance of breaking into her safe on the first attempt?

there is only one way to arrange all the eight relative to each other because they are not distinct. we have

now there are nine places to place an (and not more than one in one place).   because we have four symbols, we need to choose four of these nine places to insert an . order does not matter because the are not distinct.

thus there are 1 · c(9,4) ways, or, if we calculate it,

thus there are 126 ways to form such "words."

is there supposed to be a picture here?

• before the lesson, students attempt the assessment task individually. you then review students’

solutions and formulate questions that will them improve their work.

• during the lesson, students work collaboratively in pairs or threes, plotting graphs, completing

tables of values and deducing equations. then, based on the number of common solutions,

students link these representations.

• in a follow-up lesson, students receive your comments on the assessment task and use these to

attempt the similar task, approaching it with insights gained from the lesson.