Combinations
TG pp 259 and LM pp. 301-317
DEPARTMENT OF EDUCATION
Recall:
PRIMING ACTIVITY
How are you?
Each group should familiarize every
member by making handshake with each
other. Count the number of handshakes
made then complete the table:
DEPARTMENT OF EDUCATION
Answer
How are you?( individual assessment)
Number of persons
ii
Number of handshakes
2
____
3
____
4
____
5
____
6
____
7
____
DEPARTMENT OF EDUCATION
Answer
How are you?
Number of persons
Number of handshakes
2
1
3
3
4
6
5
10
6
15
n
____
DEPARTMENT OF EDUCATION
Answer
How are you?
Number of persons
Number of handshakes
2
1
2!/(2-2)! 2!
3
3
3!/(3-2)! 2!
4
6
4!/ (4-2)! 2!
5
10
5!/ (5-2)! 2!
6
15
6!/ (6-2)! 2!
n
____
n!/ (n-r)! r!
DEPARTMENT OF EDUCATION
Combinations- refers to the number of
ways of selecting from a set n the order is
not important.
n!
C(n,r) =
(n - r)! r!
The number of combinations of n objects
taken r at a time denoted by
𝑛
𝑛
𝑪 𝒏, 𝒓 , 𝑛𝑪𝑟 , 𝐶𝑟 𝑜𝑟 𝐶
𝑟
DEPARTMENT OF EDUCATION
Activity 1
1. In how many ways can a committee consisting of 4
members be formed from 8 people?
2. How many polygons can be possibly formed from 6
distinct points on a plane, no three of which are
collinear?
DEPARTMENT OF EDUCATION
Activity 1
1. In how many ways can a committee consisting of 4
members be formed from 8 people?
C= 8!/(8-4)!4!= 70
2. How many polygons can be possibly formed from 6 distinct
points on a plane, no three of which are collinear?
The polygon may have 3, 4, 5, or 6 vertices. Thus the number
N possible polygons is:
𝑁 = 𝐶 6,3 + 𝐶 6,4 + 𝐶 6,5 + 𝐶 6,6
6∙5∙4
6∙5∙4∙3
6∙5∙4∙3∙2
6∙5∙4∙3∙2∙1
=
+
+
+
3!
4!
5!
6!
=5∙4+5∙3+6+1
= 20 + 15 + 7
𝑁 = 42
possible polygons
DEPARTMENT OF EDUCATION
What to Reflect on and Understand?
Activity 2: Reason Out
1. How do you determine if a situation or problem
involves permutations or combinations?
2. How do you find the number of combinations of
a. n objects taken r at a time
b. n objects when taken all at a time?
DEPARTMENT OF EDUCATION
What to Reflect on and Understand?
Activity 2: Reason Out
1. How do you determine if a situation or problem
involves permutations or combinations?
A situation or problem involves PERMUTATIONS
if the order of the objects is important, if NOT then
it is COMBINATIONS.
2. How do you find the number of combinations of
a. n objects taken r at a time
C(n,r) = n!/(n-r)!(r)!
b. n objects when taken all at a time?
C(n,n) = n!/(n-n)! (n)! = 1
DEPARTMENT OF EDUCATION
Assessment
Flex That Brain
Solve the ff. problems completely:
1. If there are12 teams in a basketball tournament
and each team must play every other team in the
eliminations, how many elimination games will
there be?
2. If there are 7 distinct points in a plane with no three
of which are collinear, how many different polygons
can be possibly formed?
3. In how many ways can a committee of 5 be formed
from 5 juniors and 7 seniors if the committee must
have 3 seniors?
DEPARTMENT OF EDUCATION
Assessment
Flex That Brain
Solve the ff. problems completely:
1. If there are12 teams in a basketball tournament
and
each team must play every other team in the
eliminations, how many elimination games will there be?
C= (12, 2) = 66 games
2. If there are 7 distinct points in a plane with no three of
which are collinear, how many different polygons can be
possibly formed?
C (total) = C(7,3) + C(7,4) +C(7,5) + C(7,6)+ C(7,7)
= 99 polygons
3. In how many ways can a committee of 5 be formed from
5 juniors and 7 seniors if the committee must have 3
seniors? C(total) = C(7,3) . C(5,2) = 350 committees
DEPARTMENT OF EDUCATION
Thank You
DEPARTMENT OF EDUCATION