Counting Ii Statistics Worksheet Page 7
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9Binomial Theorem
R and n
N,
For x, y
n
n
n
n i
i
(x + y)
=
x
y
i
i=0
The two lines above are what is called the Binomial Theorem. It gives you
an easy way to find powers of sums.
Example. We can use the Binomial Theorem and Pascal’s triangle to write
3
out the product (x + y)
. The Binomial Theorem states that
3
3
3
3 i
i
(x + y)
=
x
y
i
i=0
3
3
3
3
3 0
0
3 1
1
3 2
2
3 3
3
=
x
y
+
x
y
+
x
y
+
x
y
0
1
2
3
3
3
3
3
3
2
2
3
=
x
+
x
y +
xy
+
y
0
1
2
3
3
3
3
3
The numbers
,
,
, and
make up the fourth row of Pascal’s
0
1
2
3
triangle, and we can see from the triangle that they equal 1, 3, 3, and 1
respectively. Therefore,
3
3
2
2
3
(x + y)
= x
+ 3x
y + 3xy
+ y
Binomial coe cients. Because of the Binomial Theorem, numbers of the
n
form
are called binomial coe cients.
k
N, we can let x and y both be 1. The number 1
Example. For any n
n i
i
raised to any power equals 1, so x
= 1 and y
= 1. Also, x + y = 2. So
writing the Binomial Theorem when x = 1 and y = 1 tells us that
n
n
n
2
=
i
i=0
Notice that the equation on the above line is exactly
n
n
n
n
n
n
2
=
+
+
+ · · · +
+
0
1
2
n
1
n
47
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