SOLUTION: A lottery exists where balls numbered 1 to 18 are placed in an urn. Five balls are randomly selected. To​ win you must match all five balls in any order. How many possible outcom

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Question 1203989: A lottery exists where balls numbered 1 to 18 are placed in an urn. Five balls are randomly selected. To​ win you must match all five balls in any order. How many possible outcomes are there for this​ game?
Found 2 solutions by math_tutor2020, ikleyn:
Answer by math_tutor2020(3816)   (Show Source): You can put this solution on YOUR website!

Answer: 8568

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 = 8568

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 = 8568 different ways to play the game.

Answer by ikleyn(52776)   (Show Source): You can put this solution on YOUR website!
.

The number of all possible outcomes is the number of all possible combinations of 18 balls taken 5 at a time


     =  = 8568.    ANSWER

Solved.


------------------


In this problem, the order of 5 balls does not matter - therefore, we consider combinations.


On  Combinations,  see introductory lessons
    - Introduction to Combinations
    - PROOF of the formula on the number of Combinations
    - Problems on Combinations
in this site.



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