Refresher - Factorials, Combination, Permutation
Fundamental Principle of Counting:
Imagine putting the letters a, b, c, d into a hat, and then drawing all of them in succession. We can draw the first in 4 different ways: either a or b or c or d. After that has happened, there are 3 ways to choose the second. That is, to each of those 4 ways there correspond 3. There are 2 choices left for the next item and only one choice left for the last letter.
The number of arrangements of n different things taken n at a time is n!
By the symbol n! ("n factorial") we mean the product of consecutive numbers 1 through n. n!=n(n-1)(n-2)(n-3)⋯(3)(2)(1)
The order of the factors does not matter, whether backwards or forwards.
0! is defined as 1(0!=1)
Example 2: Five different books are on a shelf. In how many different ways could you arrange them?
Answer. 5!=5.4.3.2.1=120
An arrangement of r objects, WITHOUT regard to ORDER and without repetition, selected from n distinct objects is called a combination of n objects taken r at a time. The number of such combinations is denoted by Cnr=n!r!(n-r)!
An ORDER of arrangements of r objects, without repetition, selected from n distinct objects is called a permutation of n objects taken r at a time, and is denoted by Pnr=n!n-r! In other words, when you need to count the number of ways you can arrange items where ORDER is important, then you can use permutation to count.