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Chapter 10 Counting Techniques.

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Presentation on theme: "Chapter 10 Counting Techniques."— Presentation transcript:

1 Chapter 10 Counting Techniques

2 Permutations Section 10.2

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

4 Factorial Notation Factorial Formulas For any counting n

5 Exercises: In how many ways can 4 people be seated in a row?
4  3  2  1 = 4! = 24     If 6 horses are in a race and they all finish with no ties, in how many ways can the horses finish the race? 6  5  4  3   1 = 6! = 720 — — — — — —

6 Exercises: Formula: (n – 1)! In how many ways can 4 people
be seated in a circle? Formula: (n – 1)! (4 – 1)! = 3! = 321 = 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.

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

8 Exercise: A basketball coach must choose 4 players to
play 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

9 Assume the cards are drawn
Exercises: Assume the cards are drawn without replacement. In how many ways can 3 hearts be drawn from a standard deck of 52 cards? 13 nPr 3 = 13 x 12 x 11 = 1716 In how many ways can 2 kings be drawn from a standard deck of 52 cards? 4 nPr 2 = 4 x 3 = 12

10 Complementary Counting Principle

11 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 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 END


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