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Algebra III Lesson 45 Conditional Permutation – Two-Variable Analysis Using a Graphing Calculator Conditional Permutations Review: How are permutations written? Find: 8 P2 = 8! (8 − 2)! 8! = 6! 8 ⋅ 7 ⋅ 6! = 6! n Pr = 56 All that is meant by conditional is that maybe some outside rules apply to what kinds of combinations are allowed. There is no formula for doing these problems. You just have to adjust and modify as needed by the problem. Example 45.1 How many odd counting numbers can be formed from the digits 3, 4, and 5 if no repetition of digits is permitted? No restriction on the length of the number. 1-digit How many choices? 2 2-digit 2 2 3-digit 1 2 Total: 2 Which box should be filled first? Right How many choices? How many choices for the other box? Total: 4 Which box should be filled first? Right How many choices? 2 How many choices for the ten’s box? How many choices for the hundred’s box? Total: 4 Grand Total: 10 Example 45.2 Find the number of odd three-digit counting numbers that are less than 600? 3 items 5 10 5 What are the conditions? odd, <600 Start where? doesn’t matter, repetition okay Hundred’s box choices? 1,2,3,4,5 Ten’s box choices? 0,1,2,3,4,5,6,7,8,9 One’s box choices? 1,3,5,7,9 Total: 5·10·5 = 250 Example 45.3 Five math books and four English books are on a shelf. How many permutations are possible if the math books must be kept together and the English books must be kept together? Math-English 5 4 3 2 1 4 3 2 1 How many choices for the first spot? Total: 5!4! And… =2880 English-Math 4 3 2 1 5 4 3 2 1 How many choices for the first spot? Total: 4!5! Grand Total: 5760 And… =2880 Example 45.4 How many four-digit odd counting numbers can be formed if no repetition of digits is permitted? 4 items 8 8 7 5 Conditions? odd, no repetition Start where? one’s box, nail down that first So… Total: 8·8·7·5 =2240 Example 45.5 An elf, a gnome, a fairy, a pixie, and a leprechaun were to sit in a line. How many different ways can they sit if the elf and the gnome insist on sitting next to each other? 5 items E G Conditions? 3 2 1 elf and gnome next to each other. Place the elf somewhere (first box for ease) Gnome next. And… Total: 6 what about gnome first? Total: 6 Grand Total: 12 Two-variable analysis using a graphing calculator and any of these problems in assignments from now on. Practice a) How many three-digit counting numbers are there that are less than 300 such that all the digits are even? 3 items 1 5 Conditions? 5 all even, <300 Start where? So, 100’s, 10’s 1’s Total: 25 anywhere b) Write the equation of the following sinusoid: 10 270° 0° 90° 180° -10 y=-10sinθ 360° 3 log10 10,000 = x c) Solve for x: 4 Rewrite using power rule log10 (10,000 ) = x 3 4 Rewrite without log (10,000) 3 4 = 10 x Regroup left to a power of 10 (10 ) 3 4 4 = 10 x Clean up and solve 103 = 10x x=3 d) Find the domain of the function f ( x) = 1− 4xx From top From bottom x≥0 1 - |4x| ≠ 0 |4x| ≠ 1 So, 4x ≠ 1 Therefore, 4x ≠ -¼, ¼ ⎧ ⎨ x ∈ ℜ x ≥ 0, x ≠ ⎩ 1⎫ ⎬ 4⎭ 4x ≠ -1