Warm up!

If you flip a coin 6 times, how many different outcomes are there?

And for your exams…

…the average was a 76%…

…the median was a 84%…

…and on average, students lost CRAZY points on questions 9-17.

The Wørd

Section 8.6Counting Principles

Slips of paper with the numbers 1-20 inclusive are placed into a hat.

How many ways can you pick an odd number?

So I have only taught this…

A prime number?

A prime or an odd number?

An integer divisible by 5?

…twice in seven years of teaching.

Two integers whose sum is 8?

Slips of paper with the numbers 1-20 inclusive are placed into a hat.

Permutations

A permutation of n things is an ordering of the elements such that

ORDER MATTERS, and there is a first, second, third, and so on.

Here is time number three…

!

( )!n r

nP

n r

But there’s an easier way to think about these…

In how many different ways can a 10 question MC exam be answered

if each question has 5 answers?

…and I really hope…

There are five people waiting in line for one bathroom. In how

many different ways can they go #2?

…I don’t screw this up too

badly.

You’re at Six Flags with three friends. You’re all going to ride

the Log thingie at the same time, but you’re the only one that won’t sit in front. In how many ways can

you all sit in the log?

I’m actually a li’l jealous of Hopkins…

The state of Missouri has license plates that consist of three letters

and then three numbers. How many different license plates can

the state have?

…not only because she has much more experience

teaching this…

Distinguishable Permutations

Suppose a set of n objects has n1 of one kind of objects, n2 of a

second kind, n3 of a third and so on.The number of distinguishable

permutations is:

1 2 3

!

! ! ! ...

n

n n n

…but also because she owns Settlers of

Cattan.

In how many distinguishable ways can the letters in the word

“Google” be written?

I can’t think of a better way…

…to learn and practice

probability…In how many distinguishable ways can the letters in the word

“Parallelogram” be written?

…than by playing a board

game.Combinations

The number of combinations of n things taken r at a time is given

by: !

( )! !n r

nC

n r r

And even better yet…

Five people are running for three spots on a school committee. How

many different committees are possible?

…Settlers is not just any board

game.How many different five card poker hands are possible?

It’s a game of strategy, deceit, and

betrayal.So how do you know when to use a Permutation or a Combination?

I’m pretty good at all of those things…

8 boys and 6 girls are applying to be in Mr. Wu’s Super-Awesome Club. If he can only take 4 guys and 4 girls, how many different ways can Mr. Wu select his club

members?

…especially the deceit and betrayal.You are playing Risk, and rolling 5

dice at the same time. How many possible outcomes are there?

And there are few things in

life…How many different distinguishable permutations are

there of the word basketball?

…better than lording a Settlers win…

You are playing a game that requires you to drawr two cards

from a deck, roll one dice, and flip two coins. How many outcomes

are there?

…over one of your mortal

enemies.You are trying out for the first ever PSH co-ed cricket team. Thirty

other students are also trying out, and the roster has a limit of 16 players. How many different

teams are possible?

8.6: from the packet!And your homework!