# A test consists of 30 problems and students are told to answer any 10 of these questions. In how many different ways can they choose the 10 questions?

Feb 7, 2016

In this problem, order doesn't matter, so we use the combination formula.

#### Explanation:

Assuming C represents the number of combinations, n represents the total number of questions and r represents the number picked at a particular time, which in this case would be 10.

C = (n!)/((n - r)!(r)!)

C = (30!)/((30 - 10)!(10)!)

C = (30!)/((20!)(10!))

Using a calculator to evaluate:

C = 30 045 015

There are 30 045 015 ways of answering the test.

Note that your data (you have to pick 10 out of 30) can be represented by the notation $n {C}_{r}$ (the n should be in subscript, but the program doesn't seem to be capable of that). As I mentioned earlier, n us the total number, 30 in this case, and r is the number out of the total elements that must be picked, 10 in this case.

Practice exercises:

1. There are 99 counties in Iowa. A band wants to go through 81 counties on a tour. How many ways are there for them to go on an 81 county tour, assuming the order doesn't matter?