How many subsets are in the set 12345?

Answer 1

Number of subsets with no members = 1

Number of subsets with one member = 5.

Number of subsets with 2 members = (5 x 4)/2 = 10.

Number of subsets with 3 members = (5 x 4 x 3 /(3 x 2) = 10.

Number of subsets with 4 members = (5 x 4 x 3 x 2)/(4 x 3 x 2) = 5.

Number of subsets with 5 members = 1

Total subsets = 1 + 5 + 10 + 10 + 5 + 1= 32.

A set with n elements has 2n subsets. In this case n = 5 and 25 = 32.

The proof in the case that n = 5 uses a basic counting technique which say that if you have five things to do, multiply together the number of ways to do each step to get the total number of ways all 5 steps can be completed.

In this case you want to make a subset of {1,2,3,4,5} and the five steps consist of deciding for each of the 5 numbers whether or not to put it in the subset. At each step you have two choices: put it in or leave it out.