A = {1,2,3,4,5} How many proper subsets does A have?

Answer 1

2^5 = 32 subsets total, but subtract 1 to get the proper subsets = 31

proper means don't include the given set

5 subsets with just one element

10 with two elements

10 with three elements

5 with four elements

1 with five elements (don't count this one)

1 with no elements, the null set

32 subsets -1 = 31 proper subsets

You can also read off those numbers, 1,5,10,10,5,1 from Pascals triangle, 5th line for a 5 element set

Or, add up the combinations: 5C0 + 5C1 + 5C2 + 5C3 + 5C4 + 5C5 =1+5+10+10+5+1=32

5C0 = 5!/0!5! =1

5C1 = 5!/1!4! = 5

5C2 = 5!/2!3! =10

5C3 = 5!/3!2! =10

5C4 = 5!/4!1! = 5

5C5 = 5!/5!0! = 1