Five numbers are to be picked, without repetition, from 44

numbers to determine the winner of the Fortune Five
game in the state lottery. If the order of the numbers is
insignificant, how many different ways can a winning
quintuple be selected? What is the probability of winning?

To determine the number of different ways a winning quintuple can be selected, we need to use the concept of combinations.

First, let's define the number of available choices (n) and the number of items to be selected (r). In this case, we have n = 44 numbers and r = 5 numbers.

The formula for the total number of combinations, denoted as C(n, r), is:

C(n, r) = n! / (r! * (n - r)!)

where "!" denotes factorial (e.g., 5! = 5 x 4 x 3 x 2 x 1).

Substituting the values, we have:

C(44, 5) = 44! / (5! * (44 - 5)!)

Now, let's calculate this number:

C(44, 5) = 44! / (5! * 39!) = (44 x 43 x 42 x 41 x 40) / (5 x 4 x 3 x 2 x 1) = 1,086,008

So, there are 1,086,008 different ways to select a winning quintuple.

Now, to calculate the probability of winning, we need to know the total number of possible outcomes. In this case, since the order of the numbers is insignificant, we need to calculate the total number of combinations for choosing any 5 numbers out of 44.

Using the same formula, we get:

C(44, 5) = 1,086,008

The probability of winning is the ratio of the number of ways to win (1,086,008) to the total number of possible outcomes (1,086,008):

Probability of winning = 1,086,008 / 1,086,008 = 1

So, the probability of winning is 1, which means there is a 100% chance of winning.