Using Factorial and “M choose N” Factorial (also known as the gamma function) is a number that's defined as the product ofall positive integers less than or equal to a given integer. For example, 5! = 5 * 4 * 3 * 2 * 1= 120. The factorial can be expressed in many ways, including using an exclamation mark(5!) or using the notation n!Factorials are commonly used in probability theory because they're related to countingprobability distributions. For example, if you toss a coin three times and want to know howlikely it is that you'll get heads each time, then you can use a factorial expression like3!/(1!*2!). This means that you're counting all possible combinations of heads and tailsfrom three tosses of a coin. The theory of probability is often used in conjunction with the concept of urns. Urns are containers, which means that you can't see into the container. However, there areusually two different colors of marbles, although sometimes there are more colors. Theobjective is to draw marbles out of the container. The two methods for drawing marbles out of this container are to draw them withreplacement or without replacement. If we draw marbles of different colors with replacement, then we have two thirds probabilityof drawing a white marble and one third probability of drawing a blue marble. Since each draw is independent, the probability of drawing a white marble twice in a row is(2/3)(2/3)=4/9. However, if I am drawing without replacement, then on my first draw I have a two thirdsprobability of drawing a white marble. Since I have already drawn a white marble, myprobability of drawing one on the second draw is one half. And if I drew a blue marble onthe first draw, there would be no chance of my drawing two white marbles in a row.
The probability of drawing two white marbles in a row without replacement is two-thirdstimes 1/2, which is 2/6 or 3/9. The way you can arrange people who are in different jobs iskind of like a product that has more slots open at the beginning and fewer at the end. We have a general term for this type of arithmetic, in which we take a number and multiplyit by one smaller, then one smaller again, and so on. It's called factorial. 5! = 5 · 4 · 3 · 2 · 1 = 120 Notation five with an exclamation mark is read "five factorial," and it is equal to 5! = 5 × 4 ×3 × 2 × 1, which is 120. When we have a factorial divided by another, we can see thatmost of these numbers cancel out, because these products are equal to 1. The resultingnumber would be 7 times 6. By convention, the notation 0! (pronounced "zero factorial") is defined to equal 1. When we draw N items from a group of M items without replacement, the number ofunique groups that we can form is called m choose n. This problem, which occurs so frequently in probability, has its own symbol and notation: IfI want to know how many unique committees of five people I could form from a group often people. If you let m equal 10, and n equal 5, then there are 10 choose 5 possible permutations ofthe numbers from 1 to 10. This can be written using a special notation as 10 factorial.Divided by 10 minus 5 factorial or 5 factorial times 5 factorial. The terminology we have just discussed will be used again in subsequent discussions ofprobability. The number of teams of five people that can be formed from a group of tenpeople is a commonly occurring concept in probability.
We're interested in the number of unique teams, which is referred to as 10 to 5. It is equalto 10! ÷ 5! × 5!. Here’s a trick for quickly getting an idea of your answer when you don't have yourcalculator handy. The 10.9.8.7.6 factorial will cancel out one of the five factorials, leaving us with 5 times 4times 3 times 2 times 1. So, the sum of these products is 10/5 x 9/3 x 8/4 x 6/2 x 7, or2x3x2x7x3. This is equal to 252.