College Math

College Math Study Guide

filled by letters and then the last three positions are taken by single digits. Now, what is the total number of possibilities of different license plates be made out of this? We know that, there are 26 letters (A to Z) and a total of 10 digits (from 0 to 9). The number of ways in which first position of the license plate can be filled is 26, the second position can also be filled in 26 ways, the third in 10, the fourth in 10, and the fifth position in 10 ways. Hence, the total number of ways the license plate can be made is: 26 * 26 * 10 * 10 * 10 = 676,000 Suppose we are given a condition that the digits and letters once used cannot be repeated, then, the total number of possibilities will be different. In this case, we will have 26ways to fill the first position, but for the second position, the total number of possibilities will become (26-1) 25. Similarly, the third position can be filled in 10 ways, and fourth and fifth place can be filled in (10-1) = 9 and (9-1) = 8 ways, respectively. The total number of possibilities in this case would become: 26 * 25 * 10 * 9 * 8 = 468,000 Now, consider an example where there is a race among 5 horses. Now in howmany ways can the race be completed by all the horses, provided there is no tie. In this case, one of the horses out of five will come first, one will come second out of four, and so on. So the total number of ways will become: 5 * 4 * 3 * 2 * 1 = 120 ways This can be written in the notation of factorial as well. 5! = 5 *4*3*2*1 4.3 Permutations This section will deal with some cases of the fundamental principle of counting. Permutation can be referred to as ordering of different objects out of the set of given objects. It is similar to what we did in the horse race example above. Let us take another example. Suppose we are given six different colors and we have to design different sequences out of it; the total number of permutations will become: 6 * 5 * 4 * 3 * 2 * 1 = 720 or 6! The rule will become that the total number of permutations of n objects will be n! Let us take it to one step further. Suppose out of these colors, we have to select only 3 colors and make a sequence, then the total number of permutations will become 6 * 5 * 4 = 120

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