Pascal Triangle

1. Dr. Fe L. Faz
Dr. Rey S. Guevarra
Dr. Meredith P. Romero
Dr. Emelita D. Bautista
Mathematics Teachers
Pedro E. Diaz High School

2. CHECKING OF
ASSIGNMENT

3. Direction: Answer the
following
 How many terms are there in the expansion
of (x+y)0
 What is the second term in the expansion
of (a+i)3
 Find the indicated term in the expansion of
each given expression:
 4th term; (x + y)5
 2th term; (p + q)6
 1st term; (x + y)2

4. Problem of the day!!!
Monkey Donkey Paradox
On the first day, monkey donkey ate 1 piece of
cupcake. On the 2nd day, monkey donkey ate 1
cupcake at the morning and 1 more during nighttime
for a total of 2 cup cakes. On the third day, monkey
donkey ate 1 cup cake at the morning, 2 at lunch time
and 1 more during night time for a total of 4 cup
cakes. On the fourth day, monkey donkey ate 1 cup
cake, then 3 cup cakes and 3 more, then 1 more at the
end of the day, for the total of 8 cup cakes. If this
pattern continues, how many cup cakes will monkey
donkey eat on the 5th day? On the 6th day?

6. PASCAL’S
TRIANGLE

7. BLAISE PASCAL
(1623 - 1662)
 A French mathematician, who
discovered a pattern row
known as Pascal’s Triangle of
Coefficients.

8. PASCAL TRIANGLE
 used to find the
coefficients of the
expansion of any
integral power.

9. The coefficients may be written in either
(x + y)0
(x + y)1
(x + y)2
(x + y)3
(x + y)4
(x + y)5
(x + y)6
(x + y)7
way
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1

10. (x + y)0
(x + y)1
(x + y)2
(x + y)3
1
1 1
1 2 1
1 3 3 1

11. 1 9 36 84 126 126 84 36 9 1
0
1
2
3
4
5
6
7
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
8 1 8 28 56 70 56 28 8 1
9
1 10 45 120 210 252 210 120 45 10 10 1

12. Illustrative Examples:
1.What is the fourth term
when (x + y)7 is expanded?
Solution:
8th row: 1, 7, 21, 35, 35, 21, 7, 1
x7 + 7x6y + 21x5y2 + 35x4y3 + 35x3y4 + 21x2y5 + 7xy6 + y7
4th term: 35x4y3

13. 2. What is the third term in
the expansion of (a + i)5 ?
Solution:
6th row: 1, 5, 10, 10, 5, 1
a5 + 5a4i + 10a3i2 + 10a4i3 + 5a3i4 + i5
3rd term: 10a3i2

14. 3. What is the sum of the
numerical coefficients when
(x + y)6 is expanded?
Solution:
7th row: 1, 6, 15, 20, 15, 6, 1
1 + 6 + 15 + 20 + 15 + 6 + 1 = 64
The sum of the coefficients: 64

15. Group
Activity

16. Activity:
 Direction: Write the expanded form of each
binomial expression and identify the term asked:
1. (x + y)4; second term
2. (x + y)8; fourth term
3. (x + y)10; sixth term
4. (x + y)6; third term

17. Summary
What are the characteristics
of the product of the
binomial expression (x + y)n ,
where n represents the
integral exponent?

18. Let’s try some challenge…
Choose the letter of
the best answer.

19. The essence of
Mathematics is not to
make things
complicated but to
make complicated
things simple.

20. Agreement
Think of this:
 Add the terms in each of the
first five rows of the Pascal’s
Triangle. Compare the sum
and find a pattern for this
sequence. Make a general
formula to express this
relation.