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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?
3𝐶3 = 1
CAT, CTA, ACT, ATC, TCA, and TAC are
counted as one.
Combination of 𝑛 objects taken 𝑟
𝒏𝑪𝒓 =
𝒏!
𝒏−𝒓 !𝒓!
Combination of 𝑛 taken 𝑟 at a time where
𝑛 ≥ 𝑟 ≥0
PERMUTATION
If two letters will be formed from the letters
of the word COVID, with or without meaning,
how many arrangements are there?
𝑛𝑃𝑟 =
𝑛!
𝑛 − 𝑟 !
5𝑃2 =
5!
5 − 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?
𝑛𝐶𝑟 =
𝑛!
𝑛 − 𝑟 ! 𝑟!
5𝐶2 =
5!
5 − 2 ! 2!
5𝐶2 = 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?
𝑛𝐶𝑟 =
𝑛!
𝑛 − 𝑟 ! 𝑟!
7𝐶4 =
7!
7 − 4 ! 4!
7𝐶4 =
7!
3! 4!
7𝐶4 =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?
5𝐶3 =
5!
5 − 3 ! 3!
5𝐶3 =
5!
2! 3!
5𝐶3 = 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?
8𝐶2 =
8!
8 − 2 ! 2!
8𝐶2 =
8!
6! 2!
8𝐶2 = 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?
6𝐶3 ∙ 4𝐶2
20 ∙ 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?
6𝐶3 ∙ 4𝐶2 + 6𝐶4 ∙ 4𝐶1 + 6𝐶5 ∙ 4𝐶0
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?
7𝐶5 ∙ 6𝐶3
21 ∙ 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?
7𝐶4 ∙ 6𝐶4
35 ∙ 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?
7𝐶5 ∙ 6𝐶3 + 7𝐶4 ∙ 6𝐶4 + 7C3 ∙ 6C5 + 7C2 ∙ 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?
5𝐶1 ∙ 3𝐶1 ∙ 4𝐶1 ∙ 2𝐶1
5 ∙ 3 ∙ 4 ∙ 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?
6𝐶4
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?
6𝐶3 + 6𝐶4 + 6𝐶5 + 6𝐶6
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?
6𝐶1 + 6𝐶2 + 6𝐶3 + 6𝐶4 + 6𝐶5 + 6𝐶6
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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Combination

  • 2. Note Please prepare your scientific calculator as it will be used in our online discussion later.
  • 3. 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.
  • 4. 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:
  • 5. 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? 3𝐶3 = 1 CAT, CTA, ACT, ATC, TCA, and TAC are counted as one.
  • 6. Combination of 𝑛 objects taken 𝑟 𝒏𝑪𝒓 = 𝒏! 𝒏−𝒓 !𝒓! Combination of 𝑛 taken 𝑟 at a time where 𝑛 ≥ 𝑟 ≥0
  • 7. PERMUTATION If two letters will be formed from the letters of the word COVID, with or without meaning, how many arrangements are there? 𝑛𝑃𝑟 = 𝑛! 𝑛 − 𝑟 ! 5𝑃2 = 5! 5 − 2 ! 5𝑃2 =20 {CO, OC, CV, VC, CI, IC, CD, DC, OV, VO, OI, IO, OD, DO, VI, IV, VD, DV, ID, DI}
  • 8. COMBINATION 1. If two letters will be formed from the letters of the word COVID how many combinations are there? 𝑛𝐶𝑟 = 𝑛! 𝑛 − 𝑟 ! 𝑟! 5𝐶2 = 5! 5 − 2 ! 2! 5𝐶2 = 10 {CO, OC, CV, VC, CI, IC, CD, DC, OV, VO, OI, IO, OD, DO, VI, IV, VD, DV, ID, DI}
  • 9. COMBINATION 2. How many groups composed of 4 persons each can be formed from 7 students? 𝑛𝐶𝑟 = 𝑛! 𝑛 − 𝑟 ! 𝑟! 7𝐶4 = 7! 7 − 4 ! 4! 7𝐶4 = 7! 3! 4! 7𝐶4 =35 There are 35 groups.
  • 10. 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? 5𝐶3 = 5! 5 − 3 ! 3! 5𝐶3 = 5! 2! 3! 5𝐶3 = 10 There are 10 ways.
  • 11. 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? 8𝐶2 = 8! 8 − 2 ! 2! 8𝐶2 = 8! 6! 2! 8𝐶2 = 28 There are 28 handshakes.
  • 12. 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? 6𝐶3 ∙ 4𝐶2 20 ∙ 6 120 ways
  • 13. 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? 6𝐶3 ∙ 4𝐶2 + 6𝐶4 ∙ 4𝐶1 + 6𝐶5 ∙ 4𝐶0 120 + 60 + 6 186 ways
  • 14. 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? 7𝐶5 ∙ 6𝐶3 21 ∙ 20 420 ways
  • 15. 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? 7𝐶4 ∙ 6𝐶4 35 ∙ 15 525 ways
  • 16. 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? 7𝐶5 ∙ 6𝐶3 + 7𝐶4 ∙ 6𝐶4 + 7C3 ∙ 6C5 + 7C2 ∙ 6C6 420 + 525 + 210 + 21 1 176 ways
  • 17. 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? 5𝐶1 ∙ 3𝐶1 ∙ 4𝐶1 ∙ 2𝐶1 5 ∙ 3 ∙ 4 ∙ 2 120 ways
  • 18. 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? 6𝐶4 15 quadrilaterals
  • 19. 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? 6𝐶3 + 6𝐶4 + 6𝐶5 + 6𝐶6 20+15+6+1 42 polygons
  • 20. 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? 6𝐶1 + 6𝐶2 + 6𝐶3 + 6𝐶4 + 6𝐶5 + 6𝐶6 6 + 15 + 20 + 15 + 6 + 1 63 possible toppings
  • 21.
  • 22. 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
  • 23. 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
  • 24. 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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