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Combination

This document contains a mathematics lesson on combinations and permutations. It begins with examples of finding the number of combinations and permutations of letters. It then provides examples of combination word problems involving selecting groups from larger sets, forming committees, handshakes, and selecting exam questions. Applications of combinations in real world scenarios are also discussed, such as lotteries, fruit salads, and polygon formation using points.

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Teacher III MATHEMATICS 10 Weeks 3-4 of Quarter 3
Note Please prepare your scientific calculator as it will be used in our online discussion later.
While waiting, think of this! Below are 5 points on a plane where no three of which are collinear. How many triangles can be formed? Type your answer in the chat box (in-call messages). . . . . . A B C D E There are 10 triangles formed.
COMBINATION It is an arrangement of objects in a definite order. In permutation, order matters. It is the selection of objects from a set. In combination, order does NOT matter. PERMUTATION Recall:
PERMUTATION vs. COMBINATION How many permutations (arrangements) are there for the letters C, A, and T? 3𝑃3 = 3! 3𝑃3 = 6 {CAT, CTA, ACT, ATC, TCA, TAC} How many combinations are there for the letters C, A, and T? 3𝐢3 = 1 CAT, CTA, ACT, ATC, TCA, and TAC are counted as one.
Combination of 𝑛 objects taken π‘Ÿ 𝒏π‘ͺ𝒓 = 𝒏! π’βˆ’π’“ !𝒓! Combination of 𝑛 taken π‘Ÿ at a time where 𝑛 β‰₯ π‘Ÿ β‰₯0
PERMUTATION If two letters will be formed from the letters of the word COVID, with or without meaning, how many arrangements are there? π‘›π‘ƒπ‘Ÿ = 𝑛! 𝑛 βˆ’ π‘Ÿ ! 5𝑃2 = 5! 5 βˆ’ 2 ! 5𝑃2 =20 {CO, OC, CV, VC, CI, IC, CD, DC, OV, VO, OI, IO, OD, DO, VI, IV, VD, DV, ID, DI}
COMBINATION 1. If two letters will be formed from the letters of the word COVID how many combinations are there? π‘›πΆπ‘Ÿ = 𝑛! 𝑛 βˆ’ π‘Ÿ ! π‘Ÿ! 5𝐢2 = 5! 5 βˆ’ 2 ! 2! 5𝐢2 = 10 {CO, OC, CV, VC, CI, IC, CD, DC, OV, VO, OI, IO, OD, DO, VI, IV, VD, DV, ID, DI}
COMBINATION 2. How many groups composed of 4 persons each can be formed from 7 students? π‘›πΆπ‘Ÿ = 𝑛! 𝑛 βˆ’ π‘Ÿ ! π‘Ÿ! 7𝐢4 = 7! 7 βˆ’ 4 ! 4! 7𝐢4 = 7! 3! 4! 7𝐢4 =35 There are 35 groups.
COMBINATION 3. In an essay test, there 5 questions given where you can choose only 3 of them to answer. How many ways can you select questions to answer? 5𝐢3 = 5! 5 βˆ’ 3 ! 3! 5𝐢3 = 5! 2! 3! 5𝐢3 = 10 There are 10 ways.
COMBINATION 4. In a limited party, there are 8 persons present. If each of them shake hands exactly with one another, how many handshakes are there? 8𝐢2 = 8! 8 βˆ’ 2 ! 2! 8𝐢2 = 8! 6! 2! 8𝐢2 = 28 There are 28 handshakes.
Other Problems Involving Combinations 1. In how many ways a group, consisting of 3 boys and 2 girls, can be formed from 6 boys and 4 girls? 6𝐢3 βˆ™ 4𝐢2 20 βˆ™ 6 120 ways
Other Problems Involving Combinations 2. In how many ways a group composed of 5 members can be formed from 6 boys and 4 girls if there is at least 3 boys in the group? 6𝐢3 βˆ™ 4𝐢2 + 6𝐢4 βˆ™ 4𝐢1 + 6𝐢5 βˆ™ 4𝐢0 120 + 60 + 6 186 ways
Other Problems Involving Combinations 3. From 7 Math books and 6 Science books, in how many ways can you select 5 Math and 3 Science books to buy if all the said books are equally necessary? 7𝐢5 βˆ™ 6𝐢3 21 βˆ™ 20 420 ways
Other Problems Involving Combinations 4. From 7 Math books and 6 Science books, in how many ways can you select 8 books if the number of Math books to be bought is equal to the number of Science books? 7𝐢4 βˆ™ 6𝐢4 35 βˆ™ 15 525 ways
Other Problems Involving Combinations 5. From 7 Math books and 6 Science books, in how many ways can you select 8 books if there is at least 3 Science books to be selected? 7𝐢5 βˆ™ 6𝐢3 + 7𝐢4 βˆ™ 6𝐢4 + 7C3 βˆ™ 6C5 + 7C2 βˆ™ 6C6 420 + 525 + 210 + 21 1 176 ways
Other Problems Involving Combinations 6. In a singing contest, each contestant must perform 4 songs with different genres. If there are 5 choices for rock , 3 choices for R&B, 4 choices for pop, and 2 choices for rap, in how many ways can the contestant select her piece? 5𝐢1 βˆ™ 3𝐢1 βˆ™ 4𝐢1 βˆ™ 2𝐢1 5 βˆ™ 3 βˆ™ 4 βˆ™ 2 120 ways
Other Problems Involving Combinations 7. If there are 6 distinct points on a plane with no 3 of which are collinear, how many quadrilaterals can be possibly formed? 6𝐢4 15 quadrilaterals
Other Problems Involving Combinations 8. If there are 6 distinct points on a plane with no 3 of which are collinear, how many polygons can be possibly formed? 6𝐢3 + 6𝐢4 + 6𝐢5 + 6𝐢6 20+15+6+1 42 polygons
Other Problems Involving Combinations 9. In a pizza parlor, there are 6 different toppings, where a customer can order any number of these toppings. If you dine at the said pizza parlor, with how many possible toppings can you order your pizza? 6𝐢1 + 6𝐢2 + 6𝐢3 + 6𝐢4 + 6𝐢5 + 6𝐢6 6 + 15 + 20 + 15 + 6 + 1 63 possible toppings
Applications of Combinations 1. Selecting numbers in a lottery 7-13-26-33-35-42, 35-26-42-7-33-13 2. Selecting fruits for salad pineapple, grapes, papaya, apple, banana papaya, banana, apple, pineapple, grapes 3. Choosing members of a committee Roberto, Romina, Daniela, Marga, Cassie Marga, Daniela, Roberto, Cassie, Romina
Applications of Combinations 4. Number of handshakes Rodrigo and Leni, Leni and Rodrigo 5. Selecting problems to solve in a given list of problems Answering item 1 followed by 3 and 5 Answering first 3, then 1, and finally 3
Applications of Combinations 6. Using points on a plane to form a polygon (no three points are collinear) . . . . . . . N M O P Q S R
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