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?

1 Answer
Feb 7, 2016

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


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 #nC_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?