How many 4­letter words can be made using the letters QUIP ...

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April 24, 2013

From yesterday's practice questions... How many 4letter "words" can be made using the letters QUIP?

How many 4letter "words" can be made using the letters QUIP if Q and U must be together? NOTE: Always deal with restrictions first

and "blocked" objects are considered one item

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How many different 6letter "words" can be made from the word OTTAWA?

At first glance we might think there are 6! different "words"

But there are multiples of some letters. We are not able to distinguish between one "T" and the other "T"

Since there are 2! ways to arrange the Ts and 2! ways to arrange the As, then there will be 2! x 2! duplicates.

To account for this overcounting we simply divide

6! / (2! x 2!) or 6! / (2!2!)

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Sometimes it is more efficient to use the COMPLEMENT

(All possible arrangements) (arrangements you don't want) = (arrangements you do want)

Brooklyn has 6 different coloured sweaters to pile up in her drawers.

How many ways can she stack them so that the red sweater is not

on the bottom?

Using Direct Reasoning

TIP Deal wit the restriction

The red sweater cannot b

sweaters that can be on the

Since there are 5 other sw we know the number of arra

Answer: 5 x 5! =

Using Indirect Reasoning ( All Red sweater on botto

If there were no restriction If the Red sweater was on

Red sweater NOT on the b

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Combinations

A permutation is an arrangement of objects that need to be in a particular order. (role/job/position)

A combination is an arrangement of objects that do NOT need to be in order.

Permutation example: Choosing a President and a Vicepresident from a group of 6 people solution: P(6,2) = 6! / (62)! = 6! / 4! = 6x5 = 30 possible arrangements

Combination example: Choosing two people for a committee from a group of 6 people

In this case if we choose Bob and Julie, it would be the same as choosing Julie and Bob (order is not important = "combination") If Bob and Julie were chosen as President and VicePresident, you can see that President Bob and VP Julie is not the same as President Julie and VP Bob. (specific roles = "permutation")

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From a class of 30 students, determine the number of ways a 5 person committee can be selected to organize a class party.

a) with no restrictions Hint: Choose 5 people from a group of 30, no order

b) with Marnie on the committee HINT: Marnie must be included, so we are choosing

4 other people from the remaining 29, no order

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