A lottery is set up in which players pick six numbers from the set 1, 2, 3, ... , 39, 40. How many different ways are there to play this lottery? (In this game the order in which the numbers are picked does NOT matter.)

the number of ways to choose 6 numbers (independent of order) out of 40 is

"40 choose 6"
=C(40,6)
=40!/((40-6)!6!)
=40!/(34!6!)

To find the number of different ways to play this lottery, we need to consider the concept of combinations.

In this case, we have 40 numbers to choose from, and we need to pick 6 numbers without considering their order. This is a combination problem, specifically, "40 choose 6" or written as C(40, 6).

The formula for combinations is C(n, r) = n! / (r! * (n - r)!), where n is the total number of items, and r is the number of items to be chosen.

For this lottery, we have n = 40 (the total numbers available), and r = 6 (the number of numbers we need to pick).

Using the formula, we can calculate the number of different ways to play the lottery:

C(40, 6) = 40! / (6! * (40 - 6)!)
= 40! / (6! * 34!)

Now, let's break down the calculation further:

40! = 40 * 39 * 38 * ... * 3 * 2 * 1
6! = 6 * 5 * 4 * 3 * 2 * 1
34! = 34 * 33 * 32 * ... * 3 * 2 * 1

When we write out the entire calculation, we can simplify it:

C(40, 6) = (40 * 39 * 38 * ... * 3 * 2 * 1) / ((6 * 5 * 4 * 3 * 2 * 1) * (34 * 33 * 32 * ... * 3 * 2 * 1))

Many of the terms in the numerator and denominator cancel out, simplifying the expression:

C(40, 6) = (40 * 39 * 38 * 37 * 36 * 35) / (6 * 5 * 4 * 3 * 2 * 1)
= 8,752,875

So, there are 8,752,875 different ways to play this lottery.