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MATH 201 §A: BINOMIAL COEFFICIENTS
DUE FRIDAY 27 MAR 2009
Definition 1. For every integer n ≥ 0 and every integer k, the bi-
nomial coefficient
n k
is the coefficient of xk^ in the expansion of the
binomial power (1 + x)n.
The binomial expansion is, according to this definition,
(1) (1 + x)n^ =
∑^ n
k=
n k
xk.
Note that since (1 + x)^0 = 1 we have
0
= 1. We also have
(2)
n k
= 0 whenever k < 0 or k > n.
Our first theorem says that a table of binomial coefficients can be computed one row at a time by the “Pascal triangle” rule.
Theorem 1. For every n ≥ 1 and every k, the binomial coefficients satisfy
(3)
n k
n − 1 k − 1
n − 1 k
Proof. Let n ≥ 1 and write
(4) (1 + x)n^ = (1 + x) · (1 + x)n−^1 = (1 + x)(1 + · · · + xn−^1 ).
The coefficient of x^0 is 1 on both sides of this equation, so ( n 0
n − 1 0
n − 1 − 1
n − 1 0
by Equation (2); this proves Equation (3) in the case k = 0. The coefficient of xn^ is 1 also, proving the formula in the case k = n. For the coefficient of xk^ with 0 < k < n, rewrite Equation (4): (5)
(1+x)n^ = (1+x)
n − 1 k − 1
xk−^1 +
n − 1 k
xk^ + · · · + xn−^1
The coefficient of xk^ on the left side of Equation (5) is
(n k
. On the right side we find two ways to produce an xk^ term: either the 1 from 1
(1 + x) multiplies the term
(n− 1 k
xk, or the x from (1 + x) multiplies the term
(n− 1 k− 1
xk−^1. Thus the coefficient of xk^ on the right hand side
is
(n− 1 k− 1
(n− 1 k
Binomial coefficients can be used to enumerate subsets of finite sets.
Theorem 2. The binomial coefficient
(n k
is the number of k-element subsets of an n-element set.
Proof. In-class Exercise. Use induction on n.
Consult the proof of Theorem 6.15 in the textbook. You may also want to use the following Lemma.
Lemma 1. Let A be an n-element set and B = A ∪ {x} an n + 1- element set. If S is a k-element subset of B, then either S ⊆ A, or x ∈ S (so S − {x} has k − 1 elements) and S − {x} ⊆ A.
Proof. Since B consists of A together with one additional element x, every subset of B that does not contain x is a subset of A. Moreover, if a subset of B does contain x, then all the other elements of that subset belong to A.
Theorem 2 provides an alternate proof of the textbook’s Theorem 6.15.
Corollary. The number of subsets of an n-element set is 2 n.
Proof. By Theorem 2 and the binomial expansion Equation (1), the number of subsets of an n-element set is ∑^ n
k=
n k
= (1 + 1)n^ = 2n.
The expression
(n k
is pronounced “binomial n, k.” Because of The- orem 2 the symbol is often read as “n choose k.” Homework assignment: Theorem 3 gives the formula that expresses binomial coefficients in terms of factorials. Use induction to prove Theorem 3. You will need the convention that 0! = 1.
Theorem 3. The binomial coefficients are given by
(6)
n k
n! k!(n − k)! For the proof, use Theorem 1 to show that the set S = {n ∈ N ∪ { 0 } : Equation (6) holds for 0 ≤ k ≤ n}
is inductive.
