Difficulty: Intermediate | Prerequisites: Factorials, basic algebra, familiarity with summation notation (Sigma). If you are not comfortable computing n!, review that first.
The binomial theorem gives you a formula for expanding any expression of the form (a + b)^n without multiplying it out by hand. In Advanced Calculus I, this is foundational: you will use it to derive Taylor series, approximate functions, and work through convergence arguments for power series. It also underpins the binomial distribution in probability, so it crosses into statistics as well. If you have seen Pascal's triangle before, the binomial theorem is the algebraic machinery behind it.
The binomial theorem lets you expand (a + b)^n into a sum of (n + 1) terms, each built from a binomial coefficient C(n, r) multiplied by descending powers of a and ascending powers of b. The coefficients come from the factorial formula C(n, r) = n! / (r!(n − r)!). You will use this constantly when working with series expansions and polynomial approximations.
Binomial expression
Any algebraic expression containing exactly two terms joined by addition or subtraction, such as (a + b) or (x − y). In simple terms, "bi" means two, "nomial" means terms.
Binomial theorem
A formula that expands (a + b)^n for any non-negative integer n into a finite sum of terms involving binomial coefficients. Think of it as a shortcut that tells you every term in the expansion without multiplying brackets repeatedly.
Binomial coefficient, C(n, r)
The number of ways to choose r items from a set of n items, calculated as C(n, r) = n! / (r!(n − r)!). Also written as "n choose r" or with the stacked notation (n above r in parentheses). In simple terms, it counts combinations, and it also tells you the numerical coefficient of each term in a binomial expansion.
Factorial (n!)
The product of all positive integers from 1 up to n. So 5! = 5 × 4 × 3 × 2 × 1 = 120. By convention, 0! = 1. This is the building block of the binomial coefficient formula.
Pascal's triangle
A triangular array where each entry is the sum of the two entries directly above it. Row n of Pascal's triangle gives you all the binomial coefficients C(n, 0), C(n, 1), ..., C(n, n). Think of it as a visual lookup table for binomial coefficients.
For any non-negative integer n:
(a + b)^n = C(n, 0) · a^n · b^0 + C(n, 1) · a^(n−1) · b^1 + C(n, 2) · a^(n−2) · b^2 + ... + C(n, n) · a^0 · b^n
In sigma notation: (a + b)^n = Σ (r = 0 to n) C(n, r) · a^(n−r) · b^r
The expansion always contains exactly (n + 1) terms
Powers of a start at n and decrease by 1 each term
Powers of b start at 0 and increase by 1 each term
The sum of the exponents of a and b in every term always equals n
C(n, r) = n! / (r! · (n − r)!)
C(n, 0) = 1 and C(n, n) = 1, always (the first and last coefficients are 1)
C(n, r) = C(n, n − r), so the coefficients are symmetric
This symmetry is why Pascal's triangle reads the same forwards and backwards on each row
Using the theorem with n = 2:
r = 0: C(2, 0) · a^2 · b^0 = 1 · a^2 · 1 = a^2
r = 1: C(2, 1) · a^1 · b^1 = 2 · a · b = 2ab
r = 2: C(2, 2) · a^0 · b^2 = 1 · 1 · b^2 = b^2
Result: (a + b)^2 = a^2 + 2ab + b^2
Using the theorem with n = 3:
r = 0: C(3, 0) · a^3 · b^0 = a^3
r = 1: C(3, 1) · a^2 · b^1 = 3a^2b
r = 2: C(3, 2) · a^1 · b^2 = 3ab^2
r = 3: C(3, 3) · a^0 · b^3 = b^3
Result: (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
You do not always need the full expansion. The general term (the (r + 1)th term) is:
T(r+1) = C(n, r) · a^(n−r) · b^r
This is useful in exam questions that ask for "the coefficient of" a particular power.
Binomial expansion: (a + b)^n = Σ (r = 0 to n) C(n, r) · a^(n−r) · b^r
Binomial coefficient: C(n, r) = n! / (r! · (n − r)!)
Symmetry property: C(n, r) = C(n, n − r)
Sum of all coefficients: Setting a = b = 1 gives 2^n = Σ C(n, r), which means the coefficients in any row of Pascal's triangle sum to 2^n.
Pascal's triangle (first six rows):
Row 0: 1
Row 1: 1 1
Row 2: 1 2 1
Row 3: 1 3 3 1
Row 4: 1 4 6 4 1
Row 5: 1 5 10 10 5 1
The binomial theorem is how you derive the Taylor series expansion of (1 + x)^n, which is one of the core tools in calculus for approximating functions near a point. Engineers and physicists use these approximations constantly, for instance when linearising a nonlinear system (keeping only the first-order term of a binomial expansion).
In probability, the binomial distribution uses these same coefficients to calculate the likelihood of getting exactly k successes in n independent trials, such as predicting how many heads appear in 20 coin flips.
Students often forget that C(n, 0) = 1 and C(n, n) = 1. They are not zero. Both the first and last terms of any binomial expansion have a coefficient of 1.
Students sometimes confuse the (r + 1)th term with the rth term. The general term formula uses r starting from 0, so the "third term" corresponds to r = 2, not r = 3.
A common error is writing the powers of a and b so they do not sum to n. In every single term of the expansion, the exponent of a plus the exponent of b must equal n. If they do not, something has gone wrong.
Students sometimes apply the theorem to expressions like (a + b + c)^n. The standard binomial theorem applies only to two-term expressions. Three terms require the multinomial theorem.
⚠️ Expanding (a + b)^n by hand for small n (2, 3, 4) is a near-certainty on exams. Be able to do it quickly and without error.
⚠️ "Find the coefficient of x^k in the expansion of..." is a classic exam question. Use the general term formula T(r+1) = C(n, r) · a^(n−r) · b^r and solve for r.
⚠️ The connection between the binomial theorem and Taylor/Maclaurin series (expanding (1 + x)^n) will reappear throughout the course. Understanding this link now saves time later.
⚠️ Be prepared to compute binomial coefficients by hand using the factorial formula. Simplify before you multiply to avoid enormous numbers.
1. True or false: The expansion of (a + b)^n contains n terms.
A: False. It contains (n + 1) terms.
2. What is C(5, 2)?
A: C(5, 2) = 5! / (2! · 3!) = 120 / (2 · 6) = 10.
3. Fill in the blank: In every term of a binomial expansion of (a + b)^n, the sum of the exponents of a and b equals ___.
A: n.
4. True or false: C(n, r) = C(n, n − r).
A: True. The coefficients are symmetric.
5. What is the coefficient of the second term in the expansion of (a + b)^6?
A: C(6, 1) = 6.
Q: Expand (x + 2)^4 using the binomial theorem.
A: (x + 2)^4 = C(4,0)x^4 + C(4,1)x^3(2) + C(4,2)x^2(4) + C(4,3)x(8) + C(4,4)(16) = x^4 + 8x^3 + 24x^2 + 32x + 16.
Q: Find the coefficient of x^3 in the expansion of (2x + 1)^5.
A: The general term is C(5, r)(2x)^(5−r)(1)^r = C(5, r) · 2^(5−r) · x^(5−r). Set 5 − r = 3, so r = 2. The coefficient is C(5, 2) · 2^3 = 10 · 8 = 80.
Q: What is the sum of all binomial coefficients for n = 7?
A: 2^7 = 128. This follows from setting a = b = 1 in the binomial theorem.
Q: Why does C(n, 0) always equal 1?
A: C(n, 0) = n! / (0! · n!) = 1, because 0! is defined as 1. Conceptually, there is exactly one way to choose zero items from any set.
Q: How does the binomial theorem relate to Taylor series in calculus?
A: Setting a = 1 and b = x gives (1 + x)^n = Σ C(n, r) x^r, which is a polynomial in x. For non-integer n, this generalises to the binomial series, a key example of a Taylor expansion.
This connects directly to Taylor and Maclaurin series: the binomial expansion of (1 + x)^n is one of the first power series you encounter, and understanding its structure helps when you meet more complex series later in the course.
The binomial coefficients reappear in combinatorics and probability (the binomial distribution), so this material also bridges into any statistics component of your programme.
Pascal's triangle and the recursive property C(n, r) = C(n−1, r−1) + C(n−1, r) connect to mathematical induction proofs, which are a staple of advanced calculus courses.
binomial theorem, binomial expansion, binomial coefficient, n choose r, C(n r), Pascal's triangle, factorial, combinatorics, binomial series, Taylor series, Maclaurin series, power series, polynomial expansion, Advanced Calculus I, UCF, (a + b)^n, binomial distribution, sigma notation, general term formula