Unit 2: Sets and Combinations
(Lesson 5 ~ Problems Solving with Combinations PART 1)
ALL POSSIBLE COMBINATIONS of DISTINCT ITEMS
EXAMPLE 1: Barney’s refrigerator contains 1 apple, 1 orange, and 1 pear.
Determine the number of ways that he can select at least one piece of fruit for
a snack.
METHOD 1: Using COMBINATIONS
METHOD 2: Using SUBSETS
Each subset is one combination of elements.
For a set of n elements, there are 2n subsets including the null set.
The total number of combinations containing at least one item
chosen from a group of n distinct items is 2n – 1.
EXAMPLE 2: Determine the number of ways a committee with at least 1
member can be appointed from a board with 6 members.
ALL POSSIBLE COMBINATIONS with SOME IDENTICAL ITEMS
In a situation where it is possible to choose all, some, or none
of the n items available, there are (n+1) choices.
If at least one item is chosen, the total number of selections that
can be made from p items of one kind, q items of another kind,
r items of another kind, and so on is:
(p + 1)(q + 1)(r + 1) … – 1
EXAMPLE 3: A gym locker contains 6 volleyballs, 3 basketballs, 5 tennis balls,
and 2 golf balls. Determine the number of ways Joe can select at least 1
sport ball for gym class.
EXAMPLE 4: A cookie jar contains 5 peanut butter, 6 chocolate chip, and 3
Oreo cookies. Determine the number of ways the cookie monster can select
at least one (some) cookie.