Section 9.1b…Section 9.1b…

CombinationsCombinations

What are they???What are they???

With permutations, we take n objects, r at a time,and different orderings of these objects areconsidered different permutations.

In some situations, different orderings don’t matter!!!

Combinations (of n objects taken r at a time) – thenumbers of ways to select the objects, regardless ofthe order in which they are arranged.

Combination Counting FormulaCombination Counting FormulaThe number of combinations of n objects taken rat a time is denoted C and is given byn r

!

! !n r

nC

r n r

for 0 r n

If r > n, then 0n rC A note about notation:

n rC is commonly denotedn

r

Both are read:“n choose r ”

Distinguishing Combinations Distinguishing Combinations from Permutationsfrom PermutationsIn each of the following scenarios, tell whether permutations(ordered) or combinations (unordered) are being described.Then determine the possible choices in the scenario.

25 3 13,800P

1. A president, vice-president, and secretary are chosen from a 25-member garden club.

PermutationsPermutations – order matters because it matterswho gets which office.

Distinguishing Combinations Distinguishing Combinations from Permutationsfrom PermutationsIn each of the following scenarios, tell whether permutations(ordered) or combinations (unordered) are being described.Then determine the possible choices in the scenario.

12 5

12792

5C

2. A cook chooses 5 potatoes from a bag of 12 potatoes to make a potato salad.

CombinationsCombinations – the salad is the same no matterwhat order the potatoes are chosen.

Distinguishing Combinations Distinguishing Combinations from Permutationsfrom PermutationsIn each of the following scenarios, tell whether permutations(ordered) or combinations (unordered) are being described.Then determine the possible choices in the scenario.

2730 22 6.5787 10P

3. A teacher makes a seating chart for 22 students in a classroom with 30 desks.

PermutationsPermutations – a different ordering of students inthe same seats results in a different seating chart.

More Guided PracticeMore Guided PracticeIn the Miss America pageant, 51 contestants must be narroweddown to 10 finalists who will compete on national television. Inhow many possible ways can the ten finalists be selected?

51 10C

Permutations or Combinations???

51!

10! 51 10 !

ways12,777,711,870

More Guided PracticeMore Guided PracticeThe Georgia Lotto requires winners to pick 6 integers between1 and 46. The order in which you select them does not matter;indeed, the lottery tickets are always printed with the numbersin ascending order. How many different lottery tickets arepossible?

46 6C

Permutations or Combinations???

possible tickets9,366,819

More Guided PracticeMore Guided PracticeArmando’s Pizzeria offers patrons any combination of up to 10different toppings. How many different pizzas can be ordered

10 3 120C possible pizzas

(a) if we can choose any three toppings?

Order of the toppings does not matter, so:

More Guided PracticeMore Guided PracticeArmando’s Pizzeria offers patrons any combination of up to 10different toppings. How many different pizzas can be ordered

10 rC(b) if we can choose any number of toppings (0 through 10)?

We could add up all numbers of the form

for r = 0, 1,…, 10.

A quicker way: For each option (topping), we can choose eitheryes or no. By the Multiplication Principle:

2 2 2 2 2 2 2 2 2 2 1024 possiblepizzas

Formula for Counting Formula for Counting Subsets of an Subsets of an n n - set- set

There are 2 subsets of a set with nobjects (including the empty set andthe entire set).

n

More Guided PracticeMore Guided PracticeA national hamburger chain used to advertise that it fixed itshamburgers “256 ways,” since patrons could order whatevertoppings they wanted. How many toppings must have beenavailable?

We need to solve the following equation for n:

2 256n log 2 log 256n log 2 log 256n

log 256

log 2n

8n toppings