Answers:
- Problem 14) There are 1093 different ice creams possible.
- Problem 15) There are 120 ways to pick the reading list.
- Problem 16) We have 1365 combinations.
- Problem 17) There are 315 different combos of cars possible.
- Problem 18) You can go on 21,700 different combinations of rides.
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Explanations:
Problem 14
There are n = 10 toppings to choose from.
If you order r = 0 toppings, then there is nCr = 13C0 = 1 way to do this.
Use the combination formula nCr = (n!)/(r!*(n-r)!)
If you order r = 1 topping, then there are 13C1 = 13 ways to do this.
If r = 2, then we have 13C2 = 78 different combos
If r = 3, then 13C3 = 286
If r = 4, then 13C4 = 715
Add up those subtotals: 1+13+78+286+715 = 1093
Note the use of nCr instead of nPr. The order of toppings doesn't matter.
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Problem 15
Again order doesn't matter. We have n = 10 books to pick from and r = 3 selections to make.
nCr = (n!)/(r!*(n-r)!)
10C3 = (10!)/(3!*(10-3)!)
10C3 = (10*9*8*7!)/(3!*7!)
10C3 = (10*9*8)/(3!)
10C3 = (10*9*8)/(3*2*1)
10C3 = (720)/(6)
10C3 = 120
There are 120 different ways to pick the three books (from a pool of ten).
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Problem 16
Follow the same steps as problem 15. This time you'll have n = 15 and r = 4. The answer you should get is 1365
Or you could follow these steps below:
We have 15*14*13*12 = 32,760 permutations and 4! = 24 different ways to arrange any group of 4, so we have (32,760)/(24) = 1365 combinations.
nCr = (nPr)/(r!)
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Problem 17
There are 7C1 = 7 ways to pick the color and 10C2 = 45 ways to pick the two options. Therefore, there are 7*45 = 315 different cars possible.
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Problem 18
"At least 15" means "15 or more".
- If you ride on r = 15 rides, out of n = 20 total, then we have nCr = 20C15 = 15504 different combinations.
- If you ride on r = 16 rides, out of n = 20 total, then we have nCr = 20C16 = 4845 different combinations.
- If you ride on r = 17 rides, out of n = 20 total, then we have nCr = 20C17 = 1140 different combinations.
- If you ride on r = 18 rides, out of n = 20 total, then we have nCr = 20C18 = 190 different combinations.
- If you ride on r = 19 rides, out of n = 20 total, then we have nCr = 20C19 = 20 different combinations.
- If you ride on r = 20 rides, out of n = 20 total, then we have nCr = 20C20 = 1 combo only
Add up the subtotals to get
15504+4845+1140+190+20+1 = 21,700
