How do I solve the following problem using factorials or permutations?The Super bowl Committee has applications from 9 towns to host the next two Super Bowls. How many ways can they select the host...

How do I solve the following problem using factorials or permutations?

The Super bowl Committee has applications from 9 towns to host the next two Super Bowls. How many ways can they select the host if:

            (a) The town cannot host a Super Bowl two consecutive years?

            (b) The town can host a Super Bowl two consecutive years?

Asked on by catd1115

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samhouston | Middle School Teacher | (Level 1) Associate Educator

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(a)  If the town cannot host a Super Bowl two consectutive years, use the following permuation:

Use the permuation nPk, which is the permutation of n events can form k sequences without repetition.

nPk = n! / (n - k)!

If the town cannot repeat a Super Bowl two consecutive years, then...

n = 9 (the number of events, in this case, the number of towns)

k = 2 (the number of sequences without repetition)

9! / (9 - 2)!

9! / 7!

362,880 / 5,040 = 72

If the town can host a Super Bowl two consecutive years, you do not use a permutation because a permutation requires non-repetition.  Instead, use the multiplication counting principle.

According to the multiplication counting principle, if you have m ways to make the first choice and n ways to make the second choice, then you have m*n ways to make both choices.

For the first year, you have 9 choices of towns.  For the second year, you have 9 choices of towns.  Therefore...

m = 9

n = 9

m * n = 9 * 9 = 81

 

Answers:

(a)  72 ways

(b)  81 ways

 

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