Example 3. If you have five t-shirts, five pants, and two pair of shoes. How many ways you can dress up yourself? Solution: There are 3 positions to fill in. (t-shirt, pants, and shoes) t-shirt - 5 choices pants - 5 choices Therefore: There are 5 x 5 x 2 = 50 ways shoes - 2 choices Solving Problems on Permutation Using Formula To solve problems involving permutation, use the formula: P (n , r ) = n ! "read as permutation of n objects taken r at a time equals n factorial divided by quantity n minus r factorial." (Not all objects are taken at a time) SCHOU: n is the number of objects in the given problem and r is the number of objects taken at a time P(n, n) = n! " read as permutation of n objects taken n at a time equals n factorial." (All objects are taken at a time) P = (n - 1)! " read as permutation equals quantity n minus 1 factorial. (Formula for circular permutation) Example 1. How many three letter codes could be formed from the word POST? Solution: The word POST has 4 letters and we have to form a code of three letters only. Not all letters of the word POST are taken. Therefore we will use the first formula. P (n, r) = P(4, 3) = (4-3)! -11 24 - 24 = 24 ways ( 7 -7) ! 1 Example 2. How many 3-digit number can be formed from the numbers 1,2,3,4,5, and 6? Solution: There are 6 numbers given and we have to form 3-digit numbers only. Not all numbers are taken at a time. Therefore we will use the first formula. P (n, r ) = - 6! 720 720 P(6, 3) = (6-3)1 3! 6 = 120 3-digit numbers Example 3. How many ways you can arrange 7 different books in a shelf? Solution: In this problem, all of the books are taken at a time because all of these will be place in a shelf. Therefore we will use the second formula. P (n , n ) = n ! - P(7, 7) = 71 = 5,040 ways Example 4. How many ways can 5 members of the family be seated in a circular table? Solution: P = (n - 1)!- P = (5- 1)! P= 4! P = 24 ways Try these! A. Solve the given problems using fundamental counting principle. Show your solutions. 1. How many 4-digit numbers can be formed from numbers 2, 5, 8, 3, 1, and 6? 2. If you have 3 different math books, 2 different science books and 3 dictionaries. How many ways you can bring 1 of each kind in your school? B. Solve the given problems using the appropriate formula. Show your solution. 1. How many 4-digit numbers can be formed from the numbers 1 up to 9? 2. How many ways can a boutique owner display her 8 new blouses in a display cabinet? 3. How many ways can 8 PTA officers be seated in a circular table for a meeting?Lesson/ Topic: Permutation Learning Competencies: Solving problems involving permutation. ge cour9 WOL A. Definition of terms. Fundamental counting principle - states that activity A can be done in ni ways, activity B can be done in n2 ways, activity C in ne ways, and so on, then activities A, B, C can be done simultaneously in n1 x n2 X n3...Ways. Factorial of a number - is the product of all positive integers from 1 up to the given number. Factorial in symbol - ! It looks like exclamation point. Example: 1. 51 = 1 x2 x 3 x 4x 5 = 120 2. 4! = 1 x 2 x 3 x 4 = 24 If you look at the pattern at the left, it satisfies the equation 3. 3! = 1 x2 x3 = 6 n! = n(n - 1)!. 4. 2! = 1 x2 = 2 In 51, n = 5 n! = n(n - 1)! n= 1 nl= n(n-1)! 5. 1! = 1 51 = 5(5 -1)! 1! = 1(1- 1)! 6. 0! = 1 5! = 5(4)! 1! = 1. 0! 5! = 5(24) 1 =01 0! = 1 5! =120 (Note: any number multiplied by 1 is equal to the given number 1 . O! = 0!) B. Presentation of the concept and illustration Lebec Solving Problems on Permutation Using Fundamental Principle of Counting Example 1. How many 3 digit number can be formed from the digits 2, 4, 5, 6, and 8? Solution: There are 5 numbers given and we only have 3 positions to fill in since we have to form 3 digit numbers. First digit - there are 5 choices of numbers to place. Second digit - 4 choices since the first choice was place on the first position. Third digit - 3 choices since two numbers were place on the first and second. Therefore: There are 5 x 4 x 3 = 60 three digit numbers can be formed. Example 2. How many ways you can arrange 5 varieties of flowers in a row? Solution: There are 5 varieties of flowers to be arranged and all of them should be taken. Therefore there are 5 positions to fill in. 1st position - 5 choices 2nd position - 4 choices Therefore: There are 5 x 4 x 3 x 2 x 1 = 120 ways. 3rd position - 3 choices 4th position - 2 choices 5th position - 1 choice