Answer: 154
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Explanation:
I'll be using the formula
[tex]_n C _r = \frac{n!}{r!*(n-r)!}[/tex]
which is the nCr combination formula. We use nCr instead of nPr because the order of coasters doesn't matter. The whole time, the value of n stays fixed at n = 17 which is the total number of coasters to pick from.
While n is constant, the value of r will vary. It ranges from r = 15 to r = 17 inclusive of both endpoints. In other words, r will take on values from the set {15,16,17}. So we have three cases to consider. The r value is how many coasters we select. This is due to the "at least 15" which means "15 or more".
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If you ride r = 15 coasters, then we have the following number of combinations
[tex]_n C _r = \frac{n!}{r!*(n-r)!}\\\\_{17} C _{15} = \frac{17!}{15!*(17-15)!}\\\\_{17} C _{15} = \frac{17!}{15!*2!}\\\\_{17} C _{15} = \frac{17*16*15!}{15!*2!}\\\\_{17} C _{15} = \frac{17*16}{2!}\\\\_{17} C _{15} = \frac{17*16}{2*1}\\\\_{17} C _{15} = \frac{272}{2}\\\\_{17} C _{15} = 136\\\\[/tex]
There are 136 ways to ride exactly 15 coasters (when selecting from a pool of 17 total). The order doesn't matter.
We'll use this result later, so let x = 136.
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If r = 16, then we follow the same steps as above. You should get 17C16 = 17
There are 17 ways to ride 16 coasters where order doesn't matter. Effectively this is the same as saying "there are 17 ways of picking a coaster that you won't ride"
Let y = 17 so we can use it later
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Lastly, if r = 17, then nCr = 17C17 = 1 represents one way to ride all 17 coasters where order doesn't matter.
Let z = 1
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Add up the values of x, y, and z to get the final answer
x+y+z = 136+17+1 = 154
