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Permutations are a variation of the fundamental counting principle except that you are determining a certain number of combinations from a single set. Unlike with combinations, the order in which you choose elements from the set matters. For example, you would use permutations when making a password (123 is not the same as 321). Also, when you select an element from a set, you reduce the number of elements by 1 each time. Example You are required to create a 5 digit PIN chosenfrom the numbers 0–9, without repeating numbers. You decide to make your PIN 14932. You have 10 total options at first: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 You choose 1 as your first digit—there are 9 options left You choose 4 as your first digit—there are 8 options left You choose 9 as your first digit—there are 7 options left You choose 3 as your first digit—there are 6 options left You choose 2 as your first digit—there are 5 options left To figure out how many permutations there are for a PIN, you can use an algebraic expression with or without factorials (read further). Before we continue with these algebraic expressions, it’s first important to briefly examine factorials Factorials A factorial is represented by the exclamation mark (!) and means the number will be multiplied by all integers between itself and 1 (inclusive). In other words, if you have 5!, then the number 5 will be multiplied by 4, 3, 2, and 1. It is easiest to multiply the values in descending order, but you can also start with 1 and multiply up to 5. Do not include zero as it would make the whole thing equal zero. Oddly enough, however, the factorial of zero or 0! is equal to 1. The reason for this is that zero represents an empty set, which is always included in the possible combinations. Thus, if the set is empty, there is only one way the event will happen. Similarly, the formula for subsets (S = 2ⁿ) also proves this to be true: n = 0; 2⁰ = 1. Factorial Examples 10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 3,628,800 8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 40,320 0! = 1 Permutations Expressions Permutations are used to determine the total number of possible combinations that occur when the order of making selections from a single set matters. The equation for permutations can be expressed in two ways: For the equation without factorials, since the final value multiplied is determined by (n - r + 1), the number of values being multiplied together will match the number of choices or selections you are making from the set. Now that you have seen the two expressions for permutations, let’s return to the problem you first saw: There are 30,240 permutations for making a 5 digit PIN chosen from 0-9 without repeating numbers. The number of combinations is so large because the order matters. Permutations Practice Problems If 𝑛 = 7 and 𝑟 = 4, how many permutations are possible? If 𝑛 = 6 and 𝑟 = 2, how many permutations are possible? If 𝑛 = 11 and 𝑟 = 4, how many permutations are possible? If 𝑛 = 105 and 𝑟 = 2, how many permutations are possible? Pam chooses the numbers 57, 13, 2, 19, and 35 for a lottery ticket. For her ticket to win, those numbers must be the correct five numbers and be in the correct order. How many permutations does she have to choose from? 110 120 140 150 A suitcase has six symbols above its lock. Each button may only be pressed once and must be pressed in the correct order. How many permutations of the buttons are there to open the lock, assuming all buttons must be pressed once? 570 690 720 830 Your locker requires a numbered passcode with 4 digits, chosen from 0 to 9. If you cannot have any repeating numbers, how many possible passcodes are available? 5,040 6,080 7,140 7,980 If there are 30 possible combinations of a 2−digit password that can be made, how many elements are in the set from which the digits for the password were chosen (no repeats)? 3 4 5 6 Answers to Practice Problems 840 30 7,920 10,920 120 720 5,040 6
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