Pascal’s Triangle and the Binomial Theorem
Key Terms
- Pascal’s triangle
- Row n lists the binomial coefficients C(n,0), C(n,1), …, C(n,n); each entry is the sum of the two entries above it.
- Binomial theorem
- (a + b)n = ∑r=0n C(n,r) an−r br.
- General term
- Tr+1 = C(n,r) an−r br (the (r+1)th term; r starts at 0).
- Sum of row n
- ∑ C(n,r) = 2n (set a = b = 1 in the binomial theorem).
- Finding a specific term
- Set up Tr+1 and solve for r from the given power or condition before substituting.
- Symmetry
- C(n,r) = C(n,n−r), so the triangle is symmetric; the second-to-last term equals C(n,1) = n.
Pascal’s Triangle and Binomial Theorem Formulas
Pascal’s Rule: C(n, r) + C(n, r+1) = C(n+1, r+1)
Each entry in Pascal’s triangle equals the sum of the two entries directly above it.
Binomial Theorem:
(x + y)n = ∑r=0n C(n, r) xn−r yr
= C(n,0)xn + C(n,1)xn−1y + C(n,2)xn−2y2 + … + C(n,n)yn
General term (the (r+1)th term): Tr+1 = C(n, r) xn−r yr
Sum of binomial coefficients: C(n,0) + C(n,1) + … + C(n,n) = 2n
| Row n | Entries (C(n,0) to C(n,n)) | Row sum |
|---|---|---|
| 0 | 1 | 1 |
| 1 | 1 1 | 2 |
| 2 | 1 2 1 | 4 |
| 3 | 1 3 3 1 | 8 |
| 4 | 1 4 6 4 1 | 16 |
| 5 | 1 5 10 10 5 1 | 32 |
| 6 | 1 6 15 20 15 6 1 | 64 |
Worked Example 1 — Expand (x + 2)4
Use row 4 of Pascal’s triangle: coefficients are 1, 4, 6, 4, 1.
(x + 2)4 = C(4,0)x4 (2)0 + C(4,1)x3 (2)1 + C(4,2)x2 (2)2 + C(4,3)x1 (2)3 + C(4,4)x0 (2)4
= x4 + 8x3 + 24x2 + 32x + 16
Worked Example 2 — Find the Term in x3 of (x + 2)5
The general term is Tr+1 = C(5, r) x5−r 2r.
We need 5 − r = 3, so r = 2.
T3 = C(5, 2) x3 22 = 10 × x3 × 4 = 40x3
Why Does Pascal’s Rule Work?
Pascal’s rule states C(n, r) + C(n, r+1) = C(n+1, r+1). Here is a combinatorial proof:
Suppose you want to choose r+1 items from a set of n+1 items. Label one item as “special”. Every selection either:
• Includes the special item: Choose the remaining r items from the other n items → C(n, r) ways.
• Excludes the special item: Choose all r+1 items from the other n items → C(n, r+1) ways.
Since these cases are mutually exclusive and exhaustive: C(n+1, r+1) = C(n, r) + C(n, r+1). □
This is precisely the construction rule of Pascal’s triangle: each entry is the sum of the two entries above it.
Proof of the Binomial Theorem (Sketch)
When you expand (x + y)n, you multiply n brackets: (x+y)(x+y)…(x+y). Each term in the full expansion results from picking either x or y from each of the n brackets. A term xn−ryr arises when you pick y from exactly r of the n brackets (and x from the other n−r). The number of ways to choose which r brackets contribute y is exactly C(n, r). Hence the coefficient of xn−ryr is C(n, r).
The General Term Tr+1
The terms in the expansion of (x + y)n are numbered T1, T2, …, Tn+1. The (r+1)th term is:
Tr+1 = C(n, r) xn−r yr
where r = 0 gives T1 (the first term), r = 1 gives T2, and so on up to r = n giving Tn+1.
Important: If y is replaced by a constant times a power of x (e.g., expand (x + 3x2)5), write the general term, then collect powers of x to find the required term.
Sum of Binomial Coefficients = 2n
Setting x = y = 1 in the Binomial Theorem:
(1 + 1)n = ∑r=0n C(n, r) ⇒ 2n = C(n,0) + C(n,1) + … + C(n,n)
This has a natural combinatorial meaning: 2n counts all subsets of an n-element set (each element is either included or excluded). Every subset of size r contributes C(n, r) to the total, and summing over all r gives 2n.
Alternate Substitutions
Setting x = 1, y = −1: 0 = C(n,0) − C(n,1) + C(n,2) − …
This means the sum of even-position coefficients equals the sum of odd-position coefficients.
Mastery Practice
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Fluency
Q1 — Pascal’s Triangle Rows
Write down rows 7 and 8 of Pascal’s triangle (where row 0 is just “1”).
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Fluency
Q2 — Pascal’s Rule
Use Pascal’s rule to verify: (a) C(7, 3) + C(7, 4) = C(8, 4) (b) Find C(10, 4) given C(9, 3) = 84 and C(9, 4) = 126.
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Fluency
Q3 — Expand Using the Binomial Theorem
Expand fully: (a) (x + 1)4 (b) (a + b)3
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Fluency
Q4 — Expand (x + 2)4
Expand (x + 2)4 fully, simplifying all coefficients.
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Understanding
Q5 — Find a Specific Term
Find the term containing x3 in the expansion of (x + 2)5.
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Understanding
Q6 — Find the Constant Term
Find the constant term in the expansion of (x + 3/x)6.
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Understanding
Q7 — Sum of Coefficients
(a) Find the sum of all binomial coefficients in the expansion of (1 + x)8. (b) Verify using Pascal’s triangle: the sum of the entries in row 5 equals 25.
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Understanding
Q8 — Expand (2x − 3)4
Expand (2x − 3)4 fully.
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Problem Solving
Q9 — Find the Coefficient of x4
Find the coefficient of x4 in the expansion of (3 + 2x)7.
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Problem Solving
Q10 — Coefficient of x3 in a Product Expansion
Find the coefficient of x3 in the expansion of (1 + x)5(1 + x)3.