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10, Quarter 3 - Module 3, , ‘, , Department of Education ● Republic of the Philippines, , 1

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Lesson, , Combination of Objects, Without Repetition, , 1, What I Need to Know, After working on this lesson you should be able to:, 1. Illustrate combination of objects without repetition, 2. Determine the number of combination of objects without repetition given, the formula, , What I Know, Match the two columns. Write the letter only of the correct answer., Column A, 1. 6!, 2. 3! 4!, 3. (10 – 6)! C. 15 4. 9!/7!, 5. 10!/2! 8!, 6. 5P2, 7. 7P3, 8. 9P6, 9. 9P9, 10. 8P1, 11. 5C4, 12. 6C4, 13. 8C6, 14. 10C3, 15. 11C9, , Column B, A. 5, B. 8, D. 20, E. 24, F. 28, G. 45, H. 55, I. 72, J. 120, K. 144, L. 210, M. 720, N. 60, 480, O. 362, 880, , Now check your work. Refer to answer key page 25. Did you get a good, score?, If your score is at least 12 out of 15 items, you may skip the next activity and you can, proceed immediately to the next lesson. However, if you wish to answer all the, activities, your teacher will appreciate your effort., , 2

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What’s In, , Welcome to combination of objects! Before you start learning combination of, objects, let’s see what you had learned so far when you were in lower grades. This, activity will help you understand the lesson., DIRECTIONS: Answer the following in a separate answer sheet., A. Express the following product in simplest factorial form., 1. 1 ∙ 2 ∙ 3 ∙ … ∙ 8 ∙ 9 ∙ 10 2. 15 ∙, 14 ∙ 13 ∙ … ∙ 4 ∙ 3 ∙ 2 ∙∙ 1, 3. 9 ∙ 8 ∙ 7 ∙ 6!, 4. 54 ∙ 53 ∙ 52 ∙ 51!, 5. (𝑟) ∙ (𝑟 − 1) ∙ (𝑟 − 2) ∙ (𝑟 − 3) ∙ … ∙ 3 ∙ 2 ∙ 1, B. Express the following factorial notation in product form., 6., 7., 8., 9., 10., , 9!, 2! 4!, 6! + 3!, n!, (5n)!, , C. Evaluate the following., 11. 4! + 3! = ______________ 12. 3! 0! =_______________, !, 13., = _________________, 14., 5! – 3! + 4 = _____________, 15., P(5, 1) = _________________, Did this activity help you recall factorial notation?, Please check if your answers are correct through the Answer Key on page 25., If you got all the correct answers, very good! If not, review again your lesson on, factorials. Then you can proceed to lesson 1., , 3

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What’s New, COMBINATION OF OBJECTS, A combination is a mathematical technique that determines the number of, possible arrangements in a collection of items where the order of the selection, does not matter. In combinations, you can select the items in any order., Combinations can be confused with permutations., There are two types of combinations (remember the order does not, matter now), a. repetition is allowed; such as coins in your pocket (5, 5, 5,10,10), b. no repetition; such as lottery numbers (2, 14, 15, 27, 30, 33), COMBINATION WITHOUT REPETITION, This is how lotteries work. The numbers are drawn one at a time, and if, we have the lucky numbers (no matter what order) we win!, ∙, , What Is It, Let’s explore, Consider the letters L, O, V, E and take three letters at a time without, repeating any of the letters., a) if order is important, b) if order is NOT important, , Solutions:, a) If order is significant, we have, , 24 possible arrangements, LOV, LVO, OVL, OLV, VLO, VOL, , LOE, LEO, OLE, OEL, ELO, EOL, , OVE, OEV, EVO, EOV, VOE, VEO, , VEL, VLE, LEV, LVE, EVL, ELV, , b) If order is not significant, we have the following groups of three, objects, LOV, LOE, OVE, VEL, (there are only 4 out of the 24 selections that are observed), 4

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In the given example above, we notice that there are 24 arrangements of, the letters in which the order is significant, while there are only four, arrangements when the order is not significant. The number of ordered, arrangements is 6 (or 3!) times that of the number of unordered, arrangements., By observation, for any 𝑟 objects taken from 𝑛, 𝑟 may permute in 𝑟! ways., Thus to get the number of combinations, we divide nPr by 𝑟!., nPr, nCr, , =, , which is similar to, , 𝑟!, 1, nCr, , = nPr ., 𝑟!, , 𝑛!, , 𝑛!, , Substituting nPr by, , we get,, (𝑛−𝑟)!, , nCr =, , 𝑟!(𝑛−𝑟)!, , The combination of 𝑛 objects taken 𝑟 at a time is given by, 𝑛!, nCr, , =, 𝑟!(𝑛−𝑟)!, , Take note that:, , Cr = C(n, r) = 𝐶𝑟𝑛 = (𝑛𝑟),, n, , but in Lesson 1 of this module we will use the first notation., , To understand the lesson better, let us study the following example:, , 5

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Solve for the following:, 1., 2., , C2 =, 9C6 =, 4, , Solutions :, 4!, , 1., , 4C2, , =, =, =, 2!(4−2)!, , 9!, , 2., , 9C6, , 9., , =, 6, , =, , =, , =, , =, , =, , !, , =, = 84, 6!(9−6)!, , More examples:, Example :3. How many combinations are possible from the letters, S,M,I,L,E if the letters are taken, a) One at a time, b) Two at a time, c) Five at a time, Solution:, 5!, , a), , 5C1, , =, 1!(5−1)!!, , =, , =, , = =5, , 5!, , b), , 5C2, , =, , =, , =, , =, , = 10, , 2!(5−2)!!, 5!, , c), , 5C5, , =, , =, , =, , = 1, , 5!(5−5)!, , Example 4. How many different committees of four people can be formed, from a pool of seven people?, Solution:, This is forming a group of four people, order is not important, out of seven, people in the pool. Thus, this is a combination problem., 6

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7!, 7C4, , =, , =, , =, , = 35, , 4!(7−4)!!, , Example 5. In how many ways can two singers and three dancers be, chosen to make a team if there are seven singers and nine dancers who are, qualified?, Solution:, For the given problem, we can separate it into two parts. The first part, is the number of ways the singer can be chosen, denoted by 7C2.. the second, part is the number of ways the dancer can be chosen, denoted by 9C3., , The team can be formed in, , 7C2 · 9C3., , ·, , =, , !!, 7!, , =, , · 9!, , 2! 5! 3! 6!, , =, , !, , !, , ·, , = 7·6·5! · 9 ·8 ·7·6! =, 2·1· 5!, , 3·2·1· 6!, , 42 · 504 = 1 764, 2, 6, , Therefore, there are 1, 764 ways of which the two singers and three dancers, can be chosen to make a team., Example 6. How many different amounts of money can be made from a, 20peso bill, a 50-peso bill, a 100–peso bill, and a 1000-peso bill?, , Solution:, Because there are 4 bills, the number of different amounts of money is, , 4C1, , 7, , + 4C2 + 4C3 + 4C4 =, , + ++, , !!!!

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=, , + ++, , !!!!, =, , 4 + 6 +4+1, , =, , 15, , Therefore, there are 15 different amounts of money made., , What I Have Learned, A. DIRECTIONS: Choose the letter of the correct answer., 1. A surveyor is tasked to survey 10 households from a stretch of 25 houses., How, many possible ways can the, surveyor, choose the, sample?, a., , b., , c., , d. 25!, !!, 2. How many different tests with 10 questions can be made from a test bank, consisting of, 20, questions?, , b., , b., , c., , d., , !!, 3. How many different 10-member committee can be formed from the senators of, the country?, , a., , b., , c., , d., , !!, 4. A bride decided to choose 2 colors from 5 of her favorite colors to be the motif, of her wedding. How many color combinations are possible?, a., 60, b. 20, c. 10, d. 5, 5. In how many ways can 3 tenors and 4 bass be selected from 6 tenors and 8, bass to perform in an opera show?, a., 201 600, b. 1400, c. 840, d.576, , B. TRUE or FALSE. Write TRUE if the statement is correct and FALSE if it is, wrong., _____6. Both combination and permutation are dependent on factorial, notation., 8

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_____7. The relationship between the combination and permutation of, objects taken at, , 𝑟, , at a time is given by the equation, , 𝑛, , C(n, r) = P(n, r) r!., , _____8. Combinations are groupings that require order, and permutations are, groupings that do not require groupings., _____9., , Each, , 𝑟-, , combination can be viewed as a subset of, , selected from a set of, , 𝑟, , elements, , 𝑛 elements., , _____10. nCn = n!, C. List all the possible combinations of the following:, 11.Alma(A), Ben (B), Carla (C), Daryl (D), Edna (E) – taken three at a time., ____________________________________________________________, ____________________________________________________________, 12.The number 5, 6, 7, 8, 9 – taken two at a time, ______________________________________________________________, ______________________________________________________________, ______________________________________________________________, 13.The symbols, @ , # , % , & , and ! - taken three at a time., ______________________________________________________________, ______________________________________________________________, ______________________________________________________________, 14.Yellow, Blue, Orange, and White – taken 2 at a time., ______________________________________________________________, ______________________________________________________________, ______________________________________________________________, 15.Letters M, O, D, U, L, E – taken 4 at a time., ______________________________________________________________, ______________________________________________________________, ______________________________________________________________, , Now check your work. Refer to answer key page 25. Did you get a good, score?, If your score is at least 9 out of 15 items, you may skip the next activity and you can, proceed immediately to the next lesson. However, if you wish to answer all the, activities, your teacher will appreciate your effort., , 9

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Assessment, A. GUIDED ASSESSMENT, DIRECTIONS: Calculate the given expressions. Select your answer from the, choices in the table below., 1, 9, 10, 15, 20, 24, , 30, , 45, , 55, , 84, , 190, , 240, , 250, , 1000, , 4,848, , 1. 2(5!) =, 6!, , 6. 6C2 =, , 11. 9C6 =, 2., , 4! =, , 7. 5C3 =, , 12. 11C9 =, , 8. 10C8 =, , 13. 10C1 =, , 9. 10C2 =, , 14. 20C2 =, , (6−1)!, , =, , 3., 5, 10!2!, , =, , 4., , 5., !, , =, , 10. 9C8 =, , 15. 20C18 =, , B. INDEPENDENT ASSESSMENT, DIRECTIONS: Answer the following problems. Write your answer on a, separate answer sheet., 1. From a group of 12 teachers, how many different committees can be, formed consisting four or five members?, 2. In how many ways can you select five red cards and two black cards from, 10 red cards and nine black cards?, , 10

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3. The Beta Club has 14 male and 16 female members., , A committee, , composed of three men and three women is to be formed. In how many, ways can this be done?, 4. How many different committees can be selected from 8 men and 10, women if a committee is composed of three men and three women?, 5. Students are to answer four out of five exam questions. In how many, different ways can the questions be selected?, 6. In how many ways can a committee of four be selected from a group of 8, people?, 7. A committee of five is selected from five men and six women. How many, committees are possible if there must be at least three men in the, committee?, 8. From a group of 5 swimmers and 8 runners, an athletic contingent of, seven is to be formed. How many teams are possible if there are two, swimmers and 5 runners?, 9. In how many ways can a research team of five members be formed from a, group of 12 scientists of which four are chemists and 8 are physicists if the, team must include exactly two chemists?, 10. The board of directors of a corporation is composed of 10 members. An, executive board with 4 members are to be selected. In how many ways is, this possible?, 11. How many chords can be formed if there are seven points on the, circumference of the circle?, 12. Venus visited a create-your-own pizza parlor. Find the number of different, pizzas she can create using these toppings –pepperoni, anchovies, bacon,, or ham if only 3 toppings are to be selected?, 13. A private scholarship program opens 4 slots for students’ financial aid. If, there are 10 students applying for the grant, in how many ways can this, scholarship be awarded?, 14. How many different 10-member committee can be formed from the, senators of the country?, 15. In how many ways can 5 basketball players and two badminton players be, selected from 9 basketball players and 8 badminton players to represent, their teams in a friendly game with another school?, , 11

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Now check your work. Refer to answer key page 26. Did you get a good, score?, If your score is at least 9 out of 15 items, you may skip the next activity and you can, proceed immediately to the next lesson. However, if you wish to answer all the, , Lesson, , Combination of Objects with, Repetition, , 2, , activities, your teacher will appreciate your effort., , COMBINATION WITH REPETITION, Any selection of r objects from A, where each object can be selected more than, once, is called a combination of n objects taken r at a time with repetition. Two, combinations with repetition are considered identical if they have the same elements, repeated the same number of times, regardless of their order., (https://sites.math.northwestern.edu), , Let us say that there are five flavors of ice cream: mango, chocolate, ube,, vanilla, and strawberry. We can have three scoops. How many variations will there, be?, Let us use letters for the flavors: {m, c, u, v, s} . Example of selections, include, a) {m, m, m} - (3 scoops of mango), b) {c, v, s } - (one each of chocolate, vanilla, and strawberry), c) {m, u, u } - (one of mango, two of ube), (And just to be clear: there are n = 5 things to choose from, and we choose r = 3 of, them. Order does not matter, and we can repeat!), , To compute for the number of combination when repetition is allowed, we will, use the formula below:, , 12

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To know all the combinations with repetition of 5 taken elements in threes, using, the formula we get 35:, , CR, , 7· 5 = 35, , 2. There are seven colors of the rainbow. How many different ways are to, choose a group of three colors if repetition is allowed?, Solution: n = 7, , r=3, , CR, 3. Refer to problem #2, we will compare the number of combinations considering, repetition and NO repetition allowed., Repetition is allowed., , No repetition is allowed, , CR, , 9∙8 ∙7∙6!, , = 35, , = 3∙2∙1∙(6), !, , =, , 84, , What have you observed about the two computations? Why is this so?, , What’s More, Let’s go beyond, A. Evaluate the following:, 14

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9!, , 6. CR(5, 3) =__________, , = _________, , 1., , 3!6!, 12!, 2., , = _________, , 7. CR(6, 6) =__________, , = _________, , 8. CR(10, 8) =__________, , 10!, 11!, 3., , 11!, (9)!, 4., , = _________, , 9. CR(15, 13) =__________, , == _________, , 10. CR(20, 17) =__________, , 3!(8)!, (9)!, 5., , 7!(2)!, B. Answer the following problems considering that repetition is allowed in each, situation., 11. A young lady decided to choose 2 flavors from 5 of her favorite flavors of, ice cream to be served on her 18th birthday. How many combinations are, possible?, a. 15, b. 30, c. 45, d. 60, 12.Laura visited a create-your-own pizza parlor. Find the number of different, pizzas she can create using these toppings –pepperoni, anchovies, bacon, or, ham if only 3 toppings are to be selected?, a. 60, b. 20, c. 10, d. 5, 13.If 4 marbles are picked randomly from a jar containing 5 blue marbles and 6, yellow marbles, in how many possible ways can it happen that at least 3 of the, marbles picked are yellow?, a. 360, b. 406, c. 460, d. 600, 14., At Litoy’s Pizza Parlor, there are 5 different toppings, where a customer, can order any number of these toppings. How many possible toppings can, you actually order your pizza?, a. 221, b. 231, c. 241, d. 251, 15., A box contains 3 black balls, 2 white balls and one red ball. In how, many ways can 4 ball be chosen?, a. 116, b.126, c. 136, d. 146, , Congratulations! You have finished your module 3 illustrating combination of, objects. Now you are going to answer the post-test. Good luck! I know you can, do, it!, 15

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Summary, This module was about illustrating combination of objects, 1. with repetition, 2. without repetition, Combination of objects without repetition has this formula, , 𝑛!, C(n, r) =, , ; where n ≥ r., , Cr =, , n, , 𝑟!(𝑛−𝑟)!, Combination of objects with repetition has this formula, , CRn,r = (𝑛+𝑟𝑟−1) = (𝑟𝑛!(+𝑛𝑟−−11))!!, , ;, , where n ≥ r., , GLOSSARY OF TERMS, combination- is a selection of items from a collection, such that (unlike, permutations) the order of the selection does not matter., combination with repetition – any selection of r objects from A, where each object, can be selected more than once., combination without repetition - is the number of one-to-one functions from a set, of k identical elements into a set of n distinct elements., Factorial – is the product of all positive integers less than or equal to a given integer, and denoted by that integer and an exclamation point., , 16

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Assessment: (Post-Test), A. Choose the correct answer. Write the letter of the correct answer on a separate, sheet., 1. The value of CR(8,2) is _____., A. 6, B. 16, C. 26, D. 36, 2. The product of (6C3) and (7C1) is _____., A. 25, B. 35, C. 70, D. 140, 3. What is the total number of diagonals that can be drawn in an octagon?, A. 15, B. 20, C. 25, D. 30, 4. How many choices of 6 pocketbooks to read can be made from a set of nine, pocketbooks?, A. 60, B. 73, C. 84, D. 108, 5. From a class of 32 girls and 18 boys, how many study groups of 3 girls and 2, boys can be formed?, A. 438,880, B. 478,880, C. 638,880 D. 758,880, 6. Lucas would like to invite 9 friends to go on a trip but has room for only 6 of, them. In how many ways can they be chosen?, A. 84, B. 80, C. 72, D. 68, 7. How many subcommittees of 5 people can be formed from a committee, consisting of 12 people?, A. 210, B. 440, C. 792, D. 880, 8. How many ways can eight outfits be chosen from eleven outfits to be, modeled?, A. 85, B. 120, C. 160, D. 165, 9. How many can five basketball players be chosen from a group of ten?, A. 252, B. 260, C. 373, D. 386, 10. In how many ways can Jade select a committee of three juniors and three, seniors from a group containing seven juniors and eight seniors?, A. 1660, B. 1760, C. 1860, D. 1960, 11. In how many ways can an examination committee of 6 be chosen from 12, teachers?, 17

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A. 621, B. 723, C. 924, D. 1260, 12. In how many ways can a committee of six be formed from a group of nine, members?, A. 80, B. 84, C. 140, D. 186, 13. How many different committees can be selected from six men and five, women if a committee is composed of three men or three women?, A. 100, B. 200, C. 300, D. 400, 14. An English Club had 17 players in their squad. There were nine male players, and 8 female players. How many different teams could they select if each, team had five male players and six female players?, A. 27, B. 532, C. 835, D. 3528, 15. In how many ways can Amara choose four books from a list of ten books?, A. 210, B. 156, C. 75, D. 40, , 18

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