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The Multiplication Principle

Prepared by E.G. Gascon

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Multiplication Rule - Concept

• When one item, person, or thing is picked from a group and placed in a specific place in an arrangement, or order, the multiplication principle is at work.

• Eating a meal at a restaurant is a good example:• If there are 3 appetizers on the menu, there are 3 ways to

select an appetizer.• Assuming that you would NOT select an appetizer for a

main course, if there are 4 main course, there are 4 ways to select the main course.

• If there are 2 desserts on the menu, there are 2 different ways to select the desserts.

That is straight forward, however the question usually is: How many differ ways can one make a meal from this menu?

SO………….

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Ways to select your meal- Visually

The first three lines to the left represent the 3 appetizers,

The blue lines the 4 main course choice with each appetizer choice,

And the green lines the 2 dessert choices with each appetizer and main course choice.

Count the number of green lines and you will have the total number of ways to select your meal. 24 ways to combine your selections

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Ways to select your meal- Mathematically

• Multiplication rule• 3 appetizers times 4 main courses times 2 desserts

• 3 * 4 * 2 = 24

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Arrangement of Items - Multiplication Rule

• Combination Lock has 5 tumblers, and on each tumbler are 10 digits.

• Each tumbler therefore has a possibility of 10 different values

• The total possible combinations of digits for the lock are

• 10 * 10 * 10 * 10 * 10 = 100000 possible combinationsNotice: each tumbler is unique. The digit used on the first tumbler is available

again for the second and so on.

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Permutations – a modification of the Multiplication Rule

• What if the combination lock was electronic and it was programmed so that once a digit was set on the first tumbler, it could NOT be used again.

• The first tumbler would have 10 digits available• The second only 9 digits• The third only 8 digits• The forth only 7 digits• The fifth only 6 digits.• Then• 10 * 9 * 8 * 7 * 6 = 30240 possible combinations• Or use the Permutation formula

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Permutation by Formula.

!( , )

( )!

10!(10,5)

(10 5)!

10!(10,5)

(5)!

10*9*8*7*6*5*4*3*2*1(10,5)

5*4*3*2*1(10,5) 10*9*8*7*6

(10,5) 30240

nP n r

n r

P

P

P

P

P

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Permutations

•Defined as the number of permutations of n elements take r at a time.

•A permutation is an arrangement, schedule, or order of a selection of (r) elements from n elements

•The order does matter. This means that each subset is unique. (AB) is not the same as (BA)

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Distinguishable Permutations

11! 11*10*9*8*7*6*5*4**3*2*1 39,916,800

What if the elements of the group of n elements are not distinguishable?

Ex. regular permutation: How many ways to arrange the letters of a word, consider each letter is unique. “Mississippi”

Ex. distinguishable permutation: How many ways to arrange the letters of a word, considering that an i, s, p, and m are indistinguishable from any other i, s, p, and M. (there are 4i’s, 4s’s, 2p’s and 1M.

1 2 3

!

!* ! !...

11!

1!*4!*4!*2!11*10*9*8*7*6*5*4*3*2*1

1*4*3*2*1*4*3*2*1*2*134,650

n

n n nMust reduce the number of possibilities

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Combinations

Defined as the number of combinations of n elements take r at a time.

A combination is an subset, group, committee, or sample of a selection of (r) elements from n elements

The order does NOT matter. This means that each (AB) is the same as (BA)

:

!

( )!* !

Formula

n n

r n r r

Notice: there is an additional factor in the denominator to reduce the duplicates.

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Combination vs. Permutation

Given a set of 5 books. Arranged on a book shelf is the

multiplication rule. 5*4**3*2*1 = 120

Arranging 3 books on the shelf from the 5 is a permutation P(5,3) = 5*4*3 = 60

Selecting a set of 3 books to take on vacation would be a combination because the set of books {a, b, c} is the same as {a, c, b} is the same as {b, a, c}, etc.

There are fewer combinations of the 5 books then there are permutation.

:

!

( )!* !

5 5!

3 (5 3)!*3!

5 5*4*3*2*1

3 2*1*3*2*1

510

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Formula

n n

r n r r

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Questions / Comments / Suggestions

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