# Pascal's Triangle and Expanding Binomial Powers

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```							                                                                                      6.3

Pascal’s Triangle and Expanding
Binomial Powers

Pascal’s triangle            Pascal’s triangle
• a triangular
arrangement of
numbers with 1 in the
first row, and 1 and 1 in
the second row
• Each number in the
succeeding rows is
the sum of the two
numbers above it in the
preceding row.

Investigate                                                                  Tools
Optional
How can you use patterns to expand a power of a binomial?                    • Computer Algebra
System (CAS)
1.
Te c h n o l o g y   Tip
a)   a   b                                                              You can use the CAS
engine of a TI-Nspire™
b) a     b                                                              CAS graphing
calculator to expand a
c)   a   b                                                              power of a binomial.
• Open a new
d) a     b                                                                calculator page.
• Press b, select
2. Reflect
3:Algebra, and then
select 3:Expand.
• Enter the power
3. Reflect
of a binomial. For
example, type
(a + b)3 and press ·.
4.                                   a    b
The expansion will be
displayed.

6.3 Pascal’s Triangle and Expanding Binomial Powers • MHR 373
Example 1
Patterns in Pascal’s Triangle
a)

b)

Solution
a)

b)

Example 2
Position of Terms in Pascal’s Triangle
tn   r
n
r
n                           r                                                                  t
n                                   r                                                      t       t
n                                       r                                              t       t       t
n                                            r                                     t       t       t       t
n                                                r                             t       t       t       t       t
n                                                    r                     t       t       t       t       t        t
n                                                        r             t       t       t       t       t       t        t

tn   r
tn   r
tn    r
t   t                                                                                                tn   r

374 MHR • Functions 11 • Chapter 6
Solution

t          t               t

t
t       t
t            t       t
t             t       t           t
t        t            t       t           t
t           t             t       t           t        t
t               t        t            t       t           t          t

Example 3
Relate Other Patterns to Pascal’s Triangle

a)

b)

c)

Solution
a)

b)

c)

Term
Number, n                Term, f(n)                First
Differences             Second
1                         1                                 Differences
2
2                         3                                          1
3
3                         6                                          1
4
4                        10

a
f n        an   bn    c.
f n               n           bn        c

6.3 Pascal’s Triangle and Expanding Binomial Powers • MHR 375
b       c

b   c
b   c
b                c
n                   f n        n        n
f            f n         f n            n

Con ne ct io ns
n
If you take Mathematics
a       b                                      n
of Data Management in
grade 12, you will see       Value of n                            (a + b)n
how Pascal’s triangle                 0       (a + b)0 = 1
and expansions of
1       (a + b)1 = a + b
(a + b)n are connected to
probability.                          2       (a + b)2 = a 2 + 2ab + b 2
3       (a + b)3 = a 3 + 3a 2b + 3ab 2 + b 3
4       (a + b)4 = a 4 + 4a 3b + 6a 2b 2 + 4ab 3 + b 4

a         b
a                                        b

Example 4
Expand a Power of a Binomial

y
a) a         b           b)       m       n           c)          x                  d)           y

Solution
a)
a                                                   b

a       b       ab          ab              ab    ab    ab
ab              ab   ab
a        ab              ab    ab    ab    ab                                ab   b

376 MHR • Functions 11 • Chapter 6
b)                                                                                                                a       m
b        n
m        n             m         n                m   n     m                                n                m       n
m              n     m  n
m             mn                mn    mn                                  mn       n

c)                                                                                                             a       x
b
x                  x                            x                           x                    x
x                           x                   x
x                x                   x                   x           x            x
y
d)                                                                                                                a
b        y
y                      y                                y                           y                        y
y                       y                                   y                       y                        y

y
y

y            y               y
y           y

Key Concepts

tn   r
tn    r
tn       r
n                                                            r
n r                        r       n

n
a    b                               n

C1                                                                                                 t                t

C2

C3

C4                                                                                                                      a   b

6.3 Pascal’s Triangle and Expanding Binomial Powers • MHR 377
A Practise
For help with questions 1 and 2, refer to                                                         7.                                             Reasoning and Proving

Example 1.                                                                                                                               Representing                  Selecting Tools

1.                                                                              1                                                                       Problem Solving
1        1
Connecting                   Reﬂecting
1        2       1                          k
1  3 3 1                                                                             Communicating
x        y
1    4 6 4 1
1       5 10 10 5 1                        a) ky                                b)               xky
c)      x yk                         d) kx y

B Connect and Apply
8.

a)                                   b)
c)                                   d)

2.                                                                                               9.
tn   r

a)                                           b)                                                  a) t             b) t             c) t                     d) t
c)                                           d)               n
10.
For help with questions 3 and 4, refer to
Example 2.                                                                                             a)                                  b)
3.                                                                                                                                                          15            6
tn    r
36        9            1                                         7
a) t         t                               b) t                 t
c) t             t                           d) ta                ta                                                                                   56
b                b

4.
c)                                  d)
tn   r
10        45                                        35
a) t             b) t                    c) t                     d) t             x
28        56
For help with questions 5 to 7, refer to
Example 3.
5.

11. Chapter Problem
a)     x                      b) y                            c)                       t                                                                     1

d)           m                e)     x            y           f)          a                                                                             1            1

6.                                                                                                                                                      1         2           1

a)       a                                   b) x                                                                                            1          3            3       1
n
c)   t                                       d)       b                a

378 MHR • Functions 11 • Chapter 6
12.                                                               17.

Hint:

C     Extend
13.                           Reasoning and Proving

Representing             Selecting Tools

Problem Solving

Connecting               Reﬂecting

Communicating

18. Math Contest

n
14.                         x

x   n

15.

A        B            C          D

19. Math Contest

x
x
16.                                                                     A        B            C          D

20. Math Contest

A        B            C          D

6.3 Pascal’s Triangle and Expanding Binomial Powers • MHR 379

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