14C - Combinations and probability

A few calculations to note

• ⁿC₀ = 1 (you can only select no objects in one way)
• ⁿC_n = 1 (you can only select all objects in one way)
• ⁿC₁ = n (there are n ways to select one object from n possible objects)
• ⁿC_{n - 1} = n (there are n ways to not select one object from n possible objects)
• ⁿC_r = ⁿC_{n - r} (this symmetry can be seen in Pascal's Triangle).

14C - VIDEO EXAMPLE 1:

14C - VIDEO EXAMPLE 2:

A netball team of 7 player is chosen from 10 available people. In how many ways can the team be selected?

Combinations including or excluding specific item(s)

In some cases we will be interested in making selection that include or exclude a specific item or items. We will use the addition and multiplication principles to assist in counting.

14C - VIDEO EXAMPLE 3:

The Uber Cup is the pinnacle of Badminton for the Female Teams event. Australia is playing a preliminary tie against New Zealand and must select a team of 6 players from 12 that have nominated. In how many ways can the team be selected if Rhonda and Amanda are both on the team?

14C - VIDEO EXAMPLE 4:

The Davis cup is the premiere Men’s team’s event for Tennis. 12 players are eligible for the Australian team. However, Nick Kyrgios and Bernard Tomic will not play if they are both selected in the team. In how many ways can a team of 5 players be selected if exactly one of them is in the team?

Combinations taken from multiple groups

In certain situations we are required to make selections from multiple groups. We will use the addition and multiplication principles to assist in counting.

14C - VIDEO EXAMPLE 5:

Outdoor Education Leaders are being selected for the Year 8 camp to the Murray River. In total 3 male and 3 female leaders must be chosen from a pool of 6 boys and 7 girls. In how many ways can this be done?

Permutations and combinations combined

We can combine our knowledge of permutations and combinations to solve problems. Often we will select a group first and then arrange them

14C - VIDEO EXAMPLE 6:

At a basketball tournament there are two pools. Pool A has 5 teams and Pool B has 6 teams. If two teams from each pool progress to the finals, determine the number of ways the teams can be placed first, second, third and fourth.

Selections of any size

When the size of the selection can take on any value we have an efficient way to calculate the number of combinations of any size:

14C - VIDEO EXAMPLE 7:

A restaurant offers a buffet of 6 different cuisines from around the world. Assuming a patron tries at least one type of cuisine, how many different combinations of food are possible?

Applications of combinations to probability

The following examples illustrate the application of combinations to probability. Previously we examined examples involving permutations here.

14C - VIDEO EXAMPLE 8:

A council committee of 6 people is to be selected from 7 women and 2 men. What is the probability that at least one man is selected for the committee?