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?
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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
(a) 72 ways
(b) 81 ways
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