The combination is mathematical process to calculate the total outcomes of an event, where the order of the event outcomes does not matter. In order to calculate the combination, the formula used is ⁿC_r = n! / r ! * ( n - r ) !, where n represents the total number of items and represents the total number of items and r represents the number of items being chosen at a time.

In order to calculate a combination, at first the factorial needed to be calculated. A factorial is the product of all positive integers equal to or less than the required number. A factorial is written the number followed by an exclamation point. For an example, to write a factorial of 6, it is written as 6!. To calculate the factorial of 6, the positive integers which are equal to or less than 6 have to be multiplied in order to get the value of 6!. So, 6! = 6 * 5 * 4 * 3 * 2 * 1 = 720. Therefore multiplying the positive numbers which are less than and equal to 6, the factorial 6 is obtained as 720.

Let's have an look at another example to write an solve the factorial of 8.

The factorial of 8 would be written as 8!. To calculate 8!, the following integers needed to be multiplied 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1, and that equals 40,320.

Combination Formula

Looking at the equation to calculate combinations, the factorials are used throughout the formula. Remember, the combinations calculation formula is ⁿC_r = n! / r! * (n - r)!, where n represents the number of items, and r represents the number of items being chosen at a time. Let's look at an example of how to calculate a combination.

There are eleven new movies out to rent this week on DVD. John wants to select three movies to watch this weekend. How many combinations of movies can he select?

In this problem, John is choosing three movies from the ten new releases. 11 would represent the n variable, and 3 would represent the r variable. So, the required equation would look like ¹¹C₃ = 11! / 3! * (11 - 3)!. = 165

The first step that needs to be done is to subtract 11 minus 3 on the bottom of this equation. 11 - 3 = 8, so the required equation looks like 11! / 3! * 8!.

Next, it is required to expand each of the factorials. 11! would equal 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 on the top, and 3! * 8! would be 3 * 2 * 1 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1. The easiest way to work this problem is to cancel out the like terms. It can be observed that here is a 8, 7, 6, 5, 4, 3, 2 and 1 on both the top and bottom of the equation. These terms can be cancelled out. The equation has 11 * 10 * 9  left on top and 3 * 2 * 1 left on bottom. From here, it is just needed to be simply multiply. 11 * 10 * 9 = 990, and 3 * 2 * 1 = 6. So, our equation is now 990 / 6.

To finish this problem, 990  should be divided by 6, and the answers is obtained as 165. John now knows that he could select 120 different combinations of new-release movies this week.

Probability

In order to calculate the probability of an event occurring, this formula will be used : number of favorable outcomes / number of total outcomes

Let's look at an example of how to calculate the probability of an event occurring. At the checkout in the DVD store, John also purchased a bag of gumballs. In the bag of gumballs, there were five red, three green, four white and eight yellow gumballs. What is the probability that John drawing at random will select a yellow gumball?

John knows that if he adds all the gumballs together, there are 20 gumballs in the bag. So, the number of total outcomes is 20. John also knows that there are eight yellow gumballs, which would represent the number of favourable outcomes. So, the probability of selecting a yellow gumball at random from the bag is 8 out of 20.

All fractions, however, must be simplified. So, both 8 and 20 will divide by 4. So, 8/20 would reduce to 2/5. John knows that probability of him selecting a yellow gumball from the bag at random is 2/5.

Probability of Combinations

In order to calculate the total number of favorable outcomes combination is uses in probability and combination is a way to calculate events where order does not matter.

Let's look at an example. In a certain state’s lottery, 48 balls numbered 1 through 48 are placed in a machine and six of them are drawn at random. If the six numbers drawn match the numbers that a player had chosen, the player wins $1,000,000. In this lottery, the order the numbers are drawn in doesn’t matter. Compute the probability that you win the million-dollar prize if you purchase a single lottery ticket.

In order to compute the probability, we need to count the total number of ways six numbers can be drawn, and the number of ways the six numbers on the player’s ticket could match the six numbers drawn from the machine. Since there is no stipulation that the numbers be in any particular order, the number of possible outcomes of the lottery drawing is

⁴⁸C₆ = 12,271,512. Of these possible outcomes, only one would match all six numbers on the player’s ticket, so the probability of winning the grand prize is:

⁶C₆ / ⁴⁸C₆ = 1/ 12271512 ≈ = 0.0000000815