4.5.1 More on Sets

Questions asking for the number of subsets containing a specific number of elements are actually combinatorics problems.

The Rule

Use the combination formula nCknCk where:

  • nn = total number of elements in the set
  • kk = number of elements in the requested subset
nCk=n!k!⋅(n−k)!nCk = \frac{n!}{k! \cdot (n - k)!}

Example

Problem: How many subsets containing only 2 elements does the set {L, U, M, B, E, R} have?

Solution: 6C2=6!2!⋅4!=6×52=156C2 = \frac{6!}{2! \cdot 4!} = \frac{6 \times 5}{2} = 15

Tip: If asked for subsets containing 2 OR 3 elements, add the two combinations: nC2+nC3nC2 + nC3

Problem Set 4.5.1

Practice these problems. Type your answer and press Enter to check:

{a,b,c} has ___ 2-element subsets
{s,l,o,p,e} has ___ 3-element subsets
{a,b,c,d} has ___ 3-element subsets
{l,i,n,e,a,r} has ___ 4-element subsets
{a,b,c,d} has ___ 2-element subsets
{t,e,x,a,s} has ___ 3-element subsets
3-element subsets of 5-element set
2-element subsets of 6-element set
4-element subsets of {m,o,n,d,a,y}
4-element subsets of {d,e,c,i,m,a,l,s}
2 or 3-element subsets of {s,q,u,a,r,e}