Respuesta :
Answer: [tex]\frac{37}{1,101,716,330}\\\\[/tex]
This is the same as writing 37/1101716330
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Explanation:
Let's say the nine winning numbers are: 1,2,3,4,5,6,7,8,9
Let's further say that if we matched everything from 1 to 8, then we'll win second prize. That ninth value can be anything but 9. So we could have something like one of these examples
- 1,2,3,4,5,6,7,8,10
- 1,2,3,4,5,6,7,8,11
- 1,2,3,4,5,6,7,8,12
As you can see, there are 46-9 = 37 different choices we can have for that last slot. In other words, there are 37 items in the set {10,11,12,...,45,46} we can pick for that last ninth slot. Note that 46-10+1 = 37. This value will form the numerator of the answer.
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Now to calculate how many ways there are to select the nine values regardless if we win or lose.
Since order doesn't matter, we use the nCr combination formula.
[tex]_{n} C _{r} = \frac{n!}{r!*(n-r)!}\\\\_{46} C _{9} = \frac{46!}{9!*(46-9)!}\\\\_{46} C _{9} = \frac{46!}{9!*37!}\\\\_{46} C _{9} = \frac{46*45*44*43*42*41*40*39*38*37!}{9!*37!}\\\\[/tex]
[tex]_{46} C _{9} = \frac{46*45*44*43*42*41*40*39*38}{9!} \ \text{ ... the ``37!'' terms cancel}\\\\_{46} C _{9} = \frac{46*45*44*43*42*41*40*39*38}{9*8*7*6*5*4*3*2*1}\\\\_{46} C _{9} = \frac{399,790,821,830,400}{362,880}\\\\_{46} C _{9} = 1,101,716,330\\\\[/tex]
There are slightly over 1 billion ways to select the nine values.
This large value is the denominator of our answer which is [tex]\frac{37}{1,101,716,330}\\\\[/tex]
The 37 up top represents the number of ways to win the second prize we're after, while the massive number in the denominator represents how many ways there are to play the game.
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Notice that in the jump from step 4 to step 5, shown in the section above, we had the "37!" terms cancel out. What this means is that the prime number 37 is nowhere to be found anywhere in the factorization of 1,101,716,330
Why is this important? Because it tells us that the fraction [tex]\frac{37}{1,101,716,330}\\\\[/tex] cannot be reduced.
Side note: That fraction converts to the decimal form 0.00000003358397 approximately and that converts to the percentage form 0.000003358397%; both of which are very very small numbers. We can see that the odds are definitely stacked against the player.
