Can you help me in my assignment, i really couldn't get it. The topic and examples is in the pictures
Solving Problems Involving Permutation and Combination In the previous lesson, you learned to derive the formula for finding the permutation and combination of n objects taken r at a time. This time, you are expected to solve problems involving permutation and combinations. For further understanding of the lesson, check the illustration below: PERMUTATIONS COMBINATIONS Order is important. - Order is not important. 0 Key words: a Key words: arrangement group schedule sample order selection In solving problems involving permutations and combinations, recall these formulas: Permutation The number of permutations of n distinct objects, taking r object at the time without repetition, is given by the formuia P(n, r) Circular Permutation The number of permutations of n distinct obiects arranged in a circle is given by (n 1)!. Permutation with Repetition The number of permutations of n distinct objects, where m are alike, m; are alike, and m: are alike, is given by Where n = n1+n2 + Combination The number of combinations of n distinct objects, taking r object at the time without repetition, is given by the formula C(l'l, r) = n! [nr)!r1 Examples: 1 . How many permutations can be made from the letters at the word WESTIANS if, a. all letters are used? b. three letters are used at a time? Solution: a. There are 8 letters and if all are used. n = 8 and r = 8. Using the formula. 11! P(n, r) = (rtr)! Pa?" 8) = (3:)! :21: = 8! = 8 (7) (6) (5) (4) (3) (2) (1) = 40 320 Therefore. there are 40 320 permutations in the word WE STIANS if all letters are used. b. Since 3 letters are used at a time, n = 8 and r = 3. Using the formula. (rtr)! B! 8! P(8' 3) = [3331 _ 5: =18} [7: {6: 1% Therefore, there are 336 permutations in the word WESTIANS if three letters are used at a time. P(n, r) = "i = 336 2. If there are 6 seats in a row, how many ways can 8 persons arranged themselves? Solution: No. of person (objects), n = 6, taken 6 seats (time), r = 6 P (6, 6) = 6! =Bx5x4x3x2x1 = 720 Therefore, there are 720 ways to arranged 6 persons in 6 seats. 3. In how many ways can a president, a treasurer and a secretary be chosen from 11 candidates? Solution: Using permutation formula: The problem involves 11 candidates taken 3 at a time. i I \"1113}: 11. 11. 11x10x9x8'f (113)! = E = s! = 990 Therefore, there are 720 possible ways to choose a president, a treasurer and a secretary from 10 candidates. 4. How many ways can the letters of the word \"QUARANTINE\" be arranged? Solution: The number of letters is 10, \"A\" occurs 2 times, \"N\" occurs 2 times. Using the formula, 2 10! _ 10x9x8x7x6x5xAx3x2'! P=_v2'x1pzt=907200 Therefore, the word \"QUARANTINE\" can be arranged in 907 200 ways. 5. How many arrangements can be made from the word ENGINEERING? Solution: Since there are 11 letters, n = 9, \"E\" occurs 3 times, \"N occurs 3 times, \"(3" occurs 2 times, and \"I" occurs 2 times. 1_1! _ (11)(10)(9)(8)(7)6)(5)Q)( _ 3312121 _ w)@3(1)2)(13029(1) _277 200 Therefore, 27? 200 arrangements can be made from the word ENGINEERING. 6. A committee of 9 is to be chosen from 12 participants. How many ways this can be done? Solution: This is a combination with 12 participants taken 9 at a time, hence C(n, r) = n! n! (\"nr)! 12! 12! 12 11 10 . C(12,9)=_=_ =c )( )( we: 9! (129)! 9! 3! ST? 3)(2)(1) 1320 T = 220 Therefore, there are 220 ways in choosing a committee of 9 from 12 participants. 7. In how many ways can 4 teachers be selected from a class of 20 sections? Solution: This is a combination problem with n = 20 and r = 4. 11! cm, r) = n! (nr)! C 2 4 20! _ 20! ( 0' )' 41(204)! _ 4! 16! _ (20)(19)(18)(17)cz) (4)(3)(2)(1)7f! = 4 345 Therefore, teachers can be selected in 4 845 ways. 8. How many triangles can be formed by joining the vertices of an octagon? Solution: n = 8, r = 3 _ 11! cm, r) _ n! (nr)! _81 _ 1 C(3' 3) = 51(83)! _ 5! 3: = (swig) Ammo = 56 9. In a basket, there are 20 fruits, seven are apples, eight are oranges, and five are grapefruit. How many ways can 4 apples, 3 oranges, and 2 grapefruits be chosen? Solution: select 4 out of 7 apples, C(7, 4). select 3 out of 8 oranges, C(B, 3). select 2 out of 5 grapefruits, C(S. 2). Using the formula: C(7, 4) -C(8, 3) - C(5, 2) _ 7! _ s! _ 5! _ (74)!\" (a3)!3! (52pm = 35 - 56 - 10 = 19 600 Therefore, there are19 600 combinations of fruits. 10. There are 8 men and 6 women in a team- How many combinations can be formed from a group of 2 men and 2 women? Solution: select 2 men out of 8, C(8, 2) select 2 women out of 6, C(6, 2) Using the formula: C(B, 2) - C(6, 2) s! 5! = (e-z)!2! ' (52)!2! = 25 ' 15 = 420 Therefore, there are 420 combinations can be formed. v I. Find the number of permutation of letters of each word. 1 . BASKETBALL 2. DIFFERENT These are the questions that I 3. STATISTICS need your \"9"? 4. ACCEPTANCE 5. LEARNING II. Solve the following problems involving permutations. 1. In how many ways can 8 people be seated in a round table? 2. In how many ways can 9 different charms be arranged on a circular bracelet? 3. How many ways can 3 students line up to purchase a new textbook? 4. If a class has 20 students. how many different arrangements can 3 students give a presentation to the class? 5. How many ways can the letters of the word MARBLES be arranged? B. In how many ways can 8 teachers be assigned to 5 sections in Grade 10 level. 7. How many 3 letters word can be formed of the letters of the word RIGHT when repetition of letters is not allowed? 8. A museum has 10 paintings by Fernando Amorsolo and wants to arrange of them on the same wall. How many ways are there to do this? 9. How many ways can 8 people be seated in a circle? 10. A building contractor is planning to develop a subdivision that consists of 6 one - storey houses. 4 two - storey houses. and 2 split level houses. In how many distinguishable ways can the houses be arranged? lll. Solve the following problems involving combinations. 1. There are 6 questions on Prince essay test. He only needs to answer 3 of them, he can choose any 3 that he wants. How many different combinations of test can Prince answer? 2. There are 6 tourists in a hotel and there are only 4 rooms available. How many different combinations will there be if the tourists booked the rooms? 3. In how many ways can a basketball coach choose 5 players on his team with 12 members? 4. Kaci needs 8 more classes to complete her degree. If she met the prerequisite for all the courses. how many ways can she take 4 classes next semester. 5. How many committees of 3 can be formed from a group of 4 students. 6. A jar contains 4 red. 3 green and 2 yellow candies. Three candles are drawn at random. Find out the number of ways of selecting the candies of different colors? 7. Marson wants to order pizza. If there are 12 choices of pizza toppings. how many different combinations can he order? 8. Five friends want to play enough game of chess to be sure that everyone plays everyone else. How many games will they have to play? 9. In how many ways can a group of 2 boys and 5 girls be made from 3 boys and 7' girls? 10. In how many ways can a committee of 6 be chosen from 6 teaching and 4 non- teaching staffs if the committee must include 3 teaching and 3 non-teaching staffs
