Binomial theorem (natural number exponents) practice questions.

Determine the coefficient of x³ in the expansion of (2 + x)⁵.

• 20
• 40
• 80
• 10

Answer: 40

The x³ term is C(5,3) × 2² × x³ = 10 × 4 × x³, so the coefficient is 40. A common error is using 2³ instead of 2², giving 80.

In the expansion of (x + 3)⁶, the coefficient of x⁴ is

• 135
• 45
• 1215
• 90

Answer: 135

The x⁴ term is C(6,4) × x⁴ × 3² = 15 × 9 × x⁴ = 135x⁴. Swapping the exponents (using 3⁴) gives the distractor 1215.

State the number of terms in the full expansion of (a + b)⁹.

• 9
• 8
• 10
• 11

Answer: 10

An expansion of (a + b)ⁿ has n + 1 terms, corresponding to r = 0, 1, …, n. Here that gives 9 + 1 = 10 terms.

The term independent of x in the expansion of (x² + 1/x)⁶ is

• 20
• 6
• 1
• 15

Answer: 15

The general term is C(6,r) x^(12−3r); setting 12 − 3r = 0 gives r = 4, so the constant term is C(6,4) = 15.

Calculate the sum of all the coefficients in the expansion of (1 + x)⁷.

• 64
• 256
• 128
• 14

Answer: 128

Substituting x = 1 makes every power of x equal to 1, so the sum of the coefficients is (1 + 1)⁷ = 2⁷ = 128.

For a natural number n ≥ 2, the binomial coefficient C(n, 2) is equal to

• n(n + 1)/2
• n(n − 1)/2
• n²/2
• n!/2

Answer: n(n − 1)/2

C(n, 2) = n!/(2!(n − 2)!) = n(n − 1)/2. The expression n(n + 1)/2 is the sum of the first n natural numbers, a frequent mix-up.