standard_description: Know and apply the Binomial Theorem for the expansion of (x + y)^(n )in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle. (The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.)
problem_activity: Problem 1
Using the digits 1, 2, and 3, write an expression to answer the following questions
How many different three-digit numbers can you make when you can use each digit more than once?
How many different three-digit numbers can you make when you can only use each digit once?
Problem 2
Your phone requires you to use five numbers for the code, but you only can use numbers 1-5 for the code.
The first combination you come up with is:
| 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|
You then realize how "insecure" this number might be since anyone could guess it. Figuring this out you wonder, "How many possibilities are there for combinations if I only use each number once?"
Problem 3
You change the phone settings so that you only need a three-digit code to open the lock.
This means you have 5 numbers to choose from and 3 positions to put those numbers in, which is represented by
$$_5P_3$$
. How many different combinations can you get if you have three positions and a choice of five numbers with no repetition?


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