A presentation on Combinations

1. Teacher III
MATHEMATICS 10
Weeks 3-4 of Quarter 3

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

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

25. Always visit our Facebook group for
announcements and updates.