The Binomial Theorem gives us a formula to expand (a + b)n without laborious multiplication, where n is a non-negative integer. It connects algebra with combinatorics through the binomial coefficients.
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The Theorem
(a + b)n = sum from r=0 to n of C(n,r) · an−r · br
Written out:
(a + b)n = C(n,0)an + C(n,1)an-1b + C(n,2)an-2b2 + … + C(n,n)bn
Here, C(n, r) = n!/(r!(n−r)!) are called binomial coefficients.
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Key Features
- There are (n + 1) terms in the expansion of (a + b)n.
- The (r + 1)th term (general term) is: Tr+1 = C(n, r) · an-r · br.
- The powers of a decrease from n to 0; powers of b increase from 0 to n.
- The sum of indices in each term is always n.
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Middle Term(s)
- If n is even, there is ONE middle term: Tn/2 + 1.
- If n is odd, there are TWO middle terms: T(n+1/2) and T(n+3/2).
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Pascal's Triangle
- The binomial coefficients follow Pascal's triangle:
- 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
Each number is the sum of the two numbers above it (Pascal's identity: C(n,r) = C(n-1,r-1) + C(n-1,r)).
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Important Results
Key formulas
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Worked Examples
Expand (1 + x)4.
= C(4,0) + C(4,1)x + C(4,2)x2 + C(4,3)x3 + C(4,4)x4
= 1 + 4x + 6x2 + 4x3 + x4.
Find the 4th term in the expansion of (2x − y)6.
T4 = T3+1 = C(6,3)(2x)3(-y)3 = 20 × 8x3 × (-y)3 = -160x3 y3.
Find the middle term in (x + 1/x)8.
n = 8 (even). Middle term = T5 = C(8,4)x4(1/x)4 = 70 × 1 = 70.
Find the term independent of x in (x + 1/x2)9.
Tr+1 = C(9,r) x9-r (x-2)r = C(9,r) x9-3r. For independence: 9 - 3r = 0, r = 3. Term = C(9,3) = 84.
Using the binomial theorem, compute (1.01)5 approximately.
(1 + 0.01)5 ≈ 1 + 5(0.01) + 10(0.01)2 = 1 + 0.05 + 0.001 = 1.051.
Show that the sum of coefficients of (1 + x)n is 2n.
Put x = 1: (1+1)n = 2n = C(n,0) + C(n,1) + … + C(n,n). Proved.
Find the coefficient of x5 in (x + 3)8.
Tr+1 = C(8,r) x8-r 3r. For x5: 8-r=5, r=3. Coefficient = C(8,3) × 33 = 56 × 27 = 1512.
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Common mistakes
- Miscounting the term number: Tr+1 corresponds to r (starting from r=0).
- Forgetting to apply the sign when b is negative — (a − b)n has (-1)r in each term.
- Not simplifying the indices properly when finding the term independent of x.
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Summary
The Binomial Theorem expands (a+b)n into (n+1) terms using binomial coefficients C(n,r). The general term Tr+1 = C(n,r)an-rbr is the key formula for finding specific terms, middle terms, and terms independent of a variable.