Some Algebra 2 students were just introduced the idea of Pascal's Triangle (the figure above) and the Binomial Theorem. Their whole lesson was about expanding binomials to powers, or finding the nth term in a binomial expansion. While this may sound rather difficult, boring, and tedious it can also be a little more interesting, and useful, if you look at how to apply it. For example, the students took a test at the end of the lesson, and there were 5 extra credit questions on it. Now the entire test is made up of multiple choice questions consisting of 4 possible answers. So, if a student didn't know any of the 5 extra credit questions, what's the chance of them getting exactly 3 of them right? Well, this question can be answered by applying what they learned.

Let's start by recognizing that you have two options for each question. You can either get the answer right, or wrong. For our binomial, we will use R for getting the answer right and W for getting the answer wrong. Since there are 5 questions, we will raise the binomial to the 5th power, and the expansion will look something like this:

(R+W)⁵=R⁵+5R⁴W+10R³W²+10R²W³+5RW⁴+W⁵

Now to find the probability of getting exactly 3 questions right (R), strictly by guessing, we look at the term with R³. So the term 10R³W² is what we're looking at. We also know that in a question with four possible solutions, there's a 25% chance of guessing the right answer, and a 75% chance of guessing the wrong answer. So we put 0.25 in for R, and 0.75 in for W. Instead of having 10R³W², we now have 10(0.25)³(0.75)². Simplifying this, we get 10(0.015625)(0.5625). Simplifying it one more time gives us 0.087890625. Changing this to a percent, we end up with 8.7890625%.

So what's the conclusion? Well the question was what percent chance is there that you'll get exactly 3 of the bonus questions right if you guess on all 5? The answer is about 8.79%.

Now for those of you who don't care about the percent chance of this, are completely confused, have fallen asleep of boredom, or are just reading because you have to, I'll leave you with this:

I've shown this to my students on multiple occasions, but the question is, what went wrong in showing that 2=1? If you don't know the answer but want to, contact me. Thanks for reading.