298
Sets, Combinatorics, and Probability
applying the Binomial theorem
example 63
Using the binomial theorem, we can write the expansion of (x − 3)
4
. To match
the form of the binomial theorem, think of this expression as (x + (−3))
4
so that b
equals −3. Remember that a negative number raised to a power is positive for an
even power, negative for an odd power. Thus
(x − 3)
4
= C(4, 0)x
4
(−3)
0
+ C(4, 1)x
3
(−3)
1
+ C(4, 2)x
2
(−3)
2
+ C(4, 3)x
1
(−3)
3
+ C(4, 4)x
0
(−3)
4
= x
4
+ 4x
3
(−3) + 6x
2
(9) + 4x(−27) + 81
= x
4
− 12x
3
+ 54x
2
− 108x + 81
PraCtiCe 40 Expand (x + 1)
5
using the binomial theorem.
■
PraCtiCe 41 What is the fifth term in the expansion of (x + y)
7
?
■
The binomial theorem tells us that term k + 1 in the expansion of (a + b)
n
is C(n, k)a
n−k
b
k
. This allows us to find individual terms in the expansion without
computing the entire expression.
By using various values for a and b in the binomial theorem, certain identities
can be obtained.
example 64
Let a = b = 1 in the binomial theorem. Then
(1 + 1)
n
= C(n, 0) + C(n, 1) + c + C(n, k) + c + C(n, n)
or
2
n
= C(n, 0) + C(n, 1) + c + C(n, k) + c + C(n, n)
(2)
This result says that the sum of all the entries in row n of Pascal’s triangle equals
2
n
. Actually, Equation (2) can be proved on its own using a combinatorial proof.
The number C(n, k), the number of ways to select k items from a set of n items, can
be thought of as the number of k-element subsets of an n-element set. The right
side of Equation (2) therefore represents the total number of all the subsets (of all
sizes) of an n-element set. But we already know that the number of such subsets
is 2
n
.