Question 1203989
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Answer: <font color=red size=4>8568</font>


Work Shown:


n = 18 balls total
r = 5 selections


Order doesn't matter, so we use the nCr combination formula.
n C r = (n!)/(r!(n-r)!)
18 C 5 = (18!)/(5!*(18-5)!)
18 C 5 = (18!)/(5!*13!)
18 C 5 = (18*17*16*15*14*13!)/(5!*13!)
18 C 5 = (18*17*16*15*14)/(5!)
18 C 5 = (18*17*16*15*14)/(5*4*3*2*1)
18 C 5 = 1028160/120
18 C 5 = <font color=red size=4>8568</font>


Another approach:


There are 18*17*16*15*14 = 1028160 permutations possible. Start at 18 and count down until we fill 5 slots. 
Divide that permutation by 5! = 5*4*3*2*1 = 120 to correct for over-counting. 
For instance, the set {1,2,3,4,5} is the same as {1,3,2,4,5} since order doesn't matter. There are 120 ways to arrange any set of five items.
This is how we arrive at 1028160/120 = <font color=red size=4>8568</font> different ways to play the game. 
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