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Lottery games let you pay $\$1$ in exchange for filling gout a ticket with six different numbers, that range from $1$ to $49$. How many ways are there to fill out the ticket?

I believe since the numbers have to be different, then we need to use combinations in the following way where we remove 1 number from the range and calculate the number of combinations for the next slot:

$\binom{49}{1} + \binom{48}{1} + \binom{47}{1} + \binom{46}{1} + \binom{45}{1} + \binom{44}{1}$

The sum of above expression will yield the number of ways to fill out the ticket. Is this correct?

lucidgold
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1 Answers1

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The question is asking for subsets of size 6 from a set of 49 elements. Simply put the answer is $\binom{49}{6}$.

If you wanted to arrange them you could, you only need 6: $49\cdot 48\cdot 47\cdot 46\cdot45\cdot44 = \frac{49!}{43!}$. But the order doesn't matter. So you must divide by $6!$: $$\frac{49!}{6!\,43!} = \binom{49}{6}.$$

Generally, when you run into an "OR" statement is when you want to add up different cases.

Em.
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  • So you are using Permutations instead of Combinations? – lucidgold Dec 08 '15 at 04:16
  • I don't use those words. But yeah, I guess you could say $P(49, 6)$, or whatever notation you use. (49 permute 6). – Em. Dec 08 '15 at 04:18
  • Okay, could you please explain why my initial answer is incorrect? I am trying to understand when to use combinations vs. permutations. – lucidgold Dec 08 '15 at 04:21
  • That's why I wrote the first sentence. In my opinion, expel the words "permutation" and "combination" from your vocabulary. You need to understand what action you are taking in order to be able to count it. I will fill in some more details to my answer momentarily. – Em. Dec 08 '15 at 04:23
  • In the lotteries that I know, it is the set of (often $6$) numbers that matters. Order is irrelevant, and the numbers are listed in increasing order. So here the answer is $\binom{49}{6}$. – André Nicolas Dec 08 '15 at 04:27
  • Awesome, thank you. I believe I understand it now. I have upvoted and accepted your answer! – lucidgold Dec 08 '15 at 05:02