MathsAges 16-18

Binomial theorem coefficients

Use combinations C(n, r) and powers to find specific coefficients in the binomial expansion of (x + k)ⁿ. Review suitability for your learners before teaching.

New variation Try interactive challenge Browse templates

Choose “Save as PDF” to keep a copy. Use your browser’s paper setting for A4 or Letter.

Worksheet StudioPractice pack · Ages 16-18

Binomial theorem coefficients

Take your time. Show your thinking.

1. Find the coefficient of x^1 in the binomial expansion of (x + 4)^3.
1 · 4 · 3 · Binomial theorem coefficie…
2. Find the coefficient of x^2 in the binomial expansion of (x + 2)^4.
2 · 2 · 4 · Binomial theorem coefficie…
3. Find the coefficient of x^2 in the binomial expansion of (x + 1)^5.
2 · 1 · 5 · Binomial theorem coefficie…
4. Find the coefficient of x^1 in the binomial expansion of (x + 6)^3.
1 · 6 · 3 · Binomial theorem coefficie…
5. Find the coefficient of x^3 in the binomial expansion of (x + 7)^5.
3 · 7 · 5 · Binomial theorem coefficie…
6. Find the coefficient of x^1 in the binomial expansion of (x + 7)^3.
1 · 7 · 3 · Binomial theorem coefficie…
7. Find the coefficient of x^1 in the binomial expansion of (x + 1)^3.
1 · 1 · 3 · Binomial theorem coefficie…
8. Find the coefficient of x^2 in the binomial expansion of (x + 2)^6.
2 · 2 · 6 · Binomial theorem coefficie…
9. Find the coefficient of x^2 in the binomial expansion of (x + 4)^3.
2 · 4 · 3 · Binomial theorem coefficie…
10. Find the coefficient of x^1 in the binomial expansion of (x + 2)^4.
1 · 2 · 4 · Binomial theorem coefficie…
11. Find the coefficient of x^2 in the binomial expansion of (x + 3)^3.
2 · 3 · 3 · Binomial theorem coefficie…
12. Find the coefficient of x^2 in the binomial expansion of (x + 4)^6.
2 · 4 · 6 · Binomial theorem coefficie…

12 questions · Seed 51 · Age guidance, not curriculum certification

View the answer key

Worksheet StudioAnswer key · Ages 16-18

Binomial theorem coefficients

Answers and reasoning. Review before teaching.

1. Find the coefficient of x^1 in the binomial expansion of (x + 4)^3.
48
By the binomial theorem, the x^1 term has coefficient C(3, 2) × 4^2 = 3 × 16 = 48.
2. Find the coefficient of x^2 in the binomial expansion of (x + 2)^4.
24
By the binomial theorem, the x^2 term has coefficient C(4, 2) × 2^2 = 6 × 4 = 24.
3. Find the coefficient of x^2 in the binomial expansion of (x + 1)^5.
10
By the binomial theorem, the x^2 term has coefficient C(5, 3) × 1^3 = 10 × 1 = 10.
4. Find the coefficient of x^1 in the binomial expansion of (x + 6)^3.
108
By the binomial theorem, the x^1 term has coefficient C(3, 2) × 6^2 = 3 × 36 = 108.
5. Find the coefficient of x^3 in the binomial expansion of (x + 7)^5.
490
By the binomial theorem, the x^3 term has coefficient C(5, 2) × 7^2 = 10 × 49 = 490.
6. Find the coefficient of x^1 in the binomial expansion of (x + 7)^3.
147
By the binomial theorem, the x^1 term has coefficient C(3, 2) × 7^2 = 3 × 49 = 147.
7. Find the coefficient of x^1 in the binomial expansion of (x + 1)^3.
3
By the binomial theorem, the x^1 term has coefficient C(3, 2) × 1^2 = 3 × 1 = 3.
8. Find the coefficient of x^2 in the binomial expansion of (x + 2)^6.
240
By the binomial theorem, the x^2 term has coefficient C(6, 4) × 2^4 = 15 × 16 = 240.
9. Find the coefficient of x^2 in the binomial expansion of (x + 4)^3.
12
By the binomial theorem, the x^2 term has coefficient C(3, 1) × 4^1 = 3 × 4 = 12.
10. Find the coefficient of x^1 in the binomial expansion of (x + 2)^4.
32
By the binomial theorem, the x^1 term has coefficient C(4, 3) × 2^3 = 4 × 8 = 32.
11. Find the coefficient of x^2 in the binomial expansion of (x + 3)^3.
9
By the binomial theorem, the x^2 term has coefficient C(3, 1) × 3^1 = 3 × 3 = 9.
12. Find the coefficient of x^2 in the binomial expansion of (x + 4)^6.
3840
By the binomial theorem, the x^2 term has coefficient C(6, 4) × 4^4 = 15 × 256 = 3840.

12 questions · Seed 51 · Age guidance, not curriculum certification

Worksheet StudioPractice pack · Ages 16-18

Binomial theorem coefficients

Take your time. Show your thinking.

1. Find the coefficient of x^1 in the binomial expansion of (x + 4)^3.
1 · 4 · 3 · Binomial theorem coefficie…
2. Find the coefficient of x^2 in the binomial expansion of (x + 2)^4.
2 · 2 · 4 · Binomial theorem coefficie…
3. Find the coefficient of x^2 in the binomial expansion of (x + 1)^5.
2 · 1 · 5 · Binomial theorem coefficie…
4. Find the coefficient of x^1 in the binomial expansion of (x + 6)^3.
1 · 6 · 3 · Binomial theorem coefficie…
5. Find the coefficient of x^3 in the binomial expansion of (x + 7)^5.
3 · 7 · 5 · Binomial theorem coefficie…
6. Find the coefficient of x^1 in the binomial expansion of (x + 7)^3.
1 · 7 · 3 · Binomial theorem coefficie…
7. Find the coefficient of x^1 in the binomial expansion of (x + 1)^3.
1 · 1 · 3 · Binomial theorem coefficie…
8. Find the coefficient of x^2 in the binomial expansion of (x + 2)^6.
2 · 2 · 6 · Binomial theorem coefficie…
9. Find the coefficient of x^2 in the binomial expansion of (x + 4)^3.
2 · 4 · 3 · Binomial theorem coefficie…
10. Find the coefficient of x^1 in the binomial expansion of (x + 2)^4.
1 · 2 · 4 · Binomial theorem coefficie…
11. Find the coefficient of x^2 in the binomial expansion of (x + 3)^3.
2 · 3 · 3 · Binomial theorem coefficie…
12. Find the coefficient of x^2 in the binomial expansion of (x + 4)^6.
2 · 4 · 6 · Binomial theorem coefficie…

12 questions · Seed 51 · Age guidance, not curriculum certification

Worksheet StudioAnswer key · Ages 16-18

Binomial theorem coefficients

Answers and reasoning. Review before teaching.

1. Find the coefficient of x^1 in the binomial expansion of (x + 4)^3.
48
By the binomial theorem, the x^1 term has coefficient C(3, 2) × 4^2 = 3 × 16 = 48.
2. Find the coefficient of x^2 in the binomial expansion of (x + 2)^4.
24
By the binomial theorem, the x^2 term has coefficient C(4, 2) × 2^2 = 6 × 4 = 24.
3. Find the coefficient of x^2 in the binomial expansion of (x + 1)^5.
10
By the binomial theorem, the x^2 term has coefficient C(5, 3) × 1^3 = 10 × 1 = 10.
4. Find the coefficient of x^1 in the binomial expansion of (x + 6)^3.
108
By the binomial theorem, the x^1 term has coefficient C(3, 2) × 6^2 = 3 × 36 = 108.
5. Find the coefficient of x^3 in the binomial expansion of (x + 7)^5.
490
By the binomial theorem, the x^3 term has coefficient C(5, 2) × 7^2 = 10 × 49 = 490.
6. Find the coefficient of x^1 in the binomial expansion of (x + 7)^3.
147
By the binomial theorem, the x^1 term has coefficient C(3, 2) × 7^2 = 3 × 49 = 147.
7. Find the coefficient of x^1 in the binomial expansion of (x + 1)^3.
3
By the binomial theorem, the x^1 term has coefficient C(3, 2) × 1^2 = 3 × 1 = 3.
8. Find the coefficient of x^2 in the binomial expansion of (x + 2)^6.
240
By the binomial theorem, the x^2 term has coefficient C(6, 4) × 2^4 = 15 × 16 = 240.
9. Find the coefficient of x^2 in the binomial expansion of (x + 4)^3.
12
By the binomial theorem, the x^2 term has coefficient C(3, 1) × 4^1 = 3 × 4 = 12.
10. Find the coefficient of x^1 in the binomial expansion of (x + 2)^4.
32
By the binomial theorem, the x^1 term has coefficient C(4, 3) × 2^3 = 4 × 8 = 32.
11. Find the coefficient of x^2 in the binomial expansion of (x + 3)^3.
9
By the binomial theorem, the x^2 term has coefficient C(3, 1) × 3^1 = 3 × 3 = 9.
12. Find the coefficient of x^2 in the binomial expansion of (x + 4)^6.
3840
By the binomial theorem, the x^2 term has coefficient C(6, 4) × 4^4 = 15 × 256 = 3840.

12 questions · Seed 51 · Age guidance, not curriculum certification