I can't figure out this probability question!?

Question

In a state lottery, you must select 5 numbers (in any order) out of 40 correctly to win top prize.

Here is the question........How many ways can 5 numbers be chosen from 40 numbers?

I thought the answer would be: 40!/(40-5)! which equals 78,960,960. But the book says the answer is 658,008? I can't figure out how they got that number, maybe a book error??? Please help!

Answer 1

The way you state the question one must select (5) balls, esentially one at a time, from (40) consecutively numbered balls. The first number picked must be one of the five winners so the probability is 5 out of 40. The second must be one of the remaining (4) so the probability is 4 out of 39. Then 3 out of 38, then 2 out of 37, then 1 out of 36. The odds, being cumulative, are 5x4x3x2x1=120 over 40x39x38x37x36=78,960,960 which is120/78,960,960 or
.0000152 to 1. This is the probability that you will satisfy the question. The actual number is 1/.00000152 or 657,895.

Answer 2

The order in which the balls are chosen doesn't matter. This is the difference between Lotto (40 balls, taken without replacement) and Little Lotto (40 balls, taken with replacement). In Little Lotto, the order not only matters, but the same number can be pulled twice!

So what you want is the ways that 40 items can be taken 5 at a time or a combination. (40!) / ((40-5)! * 5!) = 658,008

Answer 3

If you can use the same number again then the answer would be much larger 40^5.

Answer 4

You forgot to divide by 5!.
mCn = m!/(n!(m-n)!)
40C5 = 40!/(5!35!) = 658,008