STEP-BY-STEP LESSON · §9.7
Pascal’s Formula and the Binomial Theorem
Understand where binomial coefficients come from.
Before you begin
Choosing an unordered subset
C(n,r)=n!/[r!(n−r)!] counts r-element subsets of n distinct objects for integers 0≤r≤n. Divide the ordered count by r! because all orders of the same members represent one subset.
A positive integer power
For a positive integer k, aᵏ multiplies k copies of a. In a binomial term, the exponents count how many factors supplied each choice. They are not multipliers in front of the bases.
Symbols
- C(n,k)
- choose k positions from n
- n,k
- Integers n≥0 and 0≤k≤n; number of factors and number of b choices
- Σ
- Add all indicated terms, here indexed by k=0,…,n
Definitions and notation for this topic
Combinations / binomial coefficient
C(n,k)=n!/[k!(n−k)!]
For integers n≥0 and 0≤k≤n, C(n,k), also written as n above k in parentheses, counts k-element subsets of an n-element set.
Order is ignored and elements are not repeated. Dividing P(n,k) by k! removes the k! orders of every selected subset; C(n,0)=C(n,n)=1.
Separate glossary example
C(5,2)=10. The subset {A,B} is the same choice as {B,A}.
C(4,2)=6
Binomial theorem and Pascal’s identity
(a+b)^n=∑_{k=0}^{n} C(n,k)a^(n−k)b^kFor real a,b and integer n≥0, the binomial theorem expands (a+b)^n by choosing which k of the n factors contribute b.
Pascal’s identity is C(n,k)=C(n−1,k−1)+C(n−1,k) for n≥2 and 1≤k≤n−1; split subsets by whether they include a fixed element. Boundary values are C(n,0)=C(n,n)=1.
Separate glossary example
(a+b)³=a³+3a²b+3ab²+b³. Also C(5,2)=C(4,1)+C(4,2)=4+6.
Step by step
Step 1 / 6
See the product
(a+b)ⁿ is a product of n identical factors. Expansion chooses a or b from each.
Each selection produces one term before like terms are combined.
Worked example
What is the coefficient of a³b² in (a+b)⁵?
- There are five factors, and a³b² requires three a choices and two b choices.
- Choose two of the five factors to contribute b.
- C(5,2)=5·4/2=10, so the coefficient is 10.
Optional self-check
Find the coefficient of xy² in (3x+y)³ and explain both factors in your answer.
Show answer
Choose which one of three factors supplies 3x: C(3,1)=3 choices. Each supplies numerical factor 3. The coefficient is 3·3=9.