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Chapter 10

Counting Techniques

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PermutationsSection 10.2

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Permutations

A permutation is an arrangement of n objects in a specific order.

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Factorial Notation

Factorial Formulas

For any counting n

1 0!

12)-1)(n-n(n !

n

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Exercises:

4 3 2 1 = 4! = 24

6 5 4 3 2 1 = 6! = 720

In how many ways can 4 people be seated in a row?

If 6 horses are in a race and they all finish with no ties,in how many ways can the horses finish the race?

— — — — — —

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Exercises:In how many ways can 4 peoplebe seated in a circle?

Formula: (n – 1)!

(4 – 1)! = 3! = 321 = 6

Notice: The answer is not the same as standing in a row. The reason is everyone could shift one seat to the right (left) but they would still be sitting in the same order or position relative to each other.

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Permutation Formula

The number of permutations of n objects taking r objects at a time (order is important and n r).

)!(! rnn

rnP

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Exercise:

A basketball coach must choose 4 players toplay in a particular game. (The team already has a center.) In how many ways can the 4 positions be filled if the coach has 10 players who can play any position?

10 nPr 4 = 10 x 9 x 8 x 7 = 5040

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Exercises:

13 nPr 3 = 13 x 12 x 11 = 1716

4 nPr 2 = 4 x 3 = 12

Assume the cards are drawn without replacement.

♥ In how many ways can 3 hearts be drawn from a standard deck of 52 cards?

In how many ways can 2 kings be drawn from a standard deck of 52 cards?

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Complementary Counting Principle

)()()( 'AnUnAn

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Exercises: Out of 5 children, in how many ways can a family have at least 1 boy?

n(A) = n(U) – n(A' ) n(at least 1 boy) = 25 – n(no boys(all girls)) = 25 – 1 = 31

END

Out of 5 children, in how many ways can a family have at least 2 boys?

n(A) = n(U) – n(A' ) n(at least 2 boys) = 25 – [n(no boys) + n(1 boy)] = 25 – [1 + 5] = 26