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.

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Worksheet StudioPractice pack · Ages 16-18

Binomial theorem coefficients

Take your time. Show your thinking.

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

12 questions · Seed 43 · 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^4 in the binomial expansion of (x + 1)^6.
15
By the binomial theorem, the x^4 term has coefficient C(6, 2) × 1^2 = 15 × 1 = 15.
2. Find the coefficient of x^1 in the binomial expansion of (x + 3)^5.
405
By the binomial theorem, the x^1 term has coefficient C(5, 4) × 3^4 = 5 × 81 = 405.
3. Find the coefficient of x^1 in the binomial expansion of (x + 2)^3.
12
By the binomial theorem, the x^1 term has coefficient C(3, 2) × 2^2 = 3 × 4 = 12.
4. Find the coefficient of x^1 in the binomial expansion of (x + 1)^6.
6
By the binomial theorem, the x^1 term has coefficient C(6, 5) × 1^5 = 6 × 1 = 6.
5. Find the coefficient of x^4 in the binomial expansion of (x + 5)^6.
375
By the binomial theorem, the x^4 term has coefficient C(6, 2) × 5^2 = 15 × 25 = 375.
6. Find the coefficient of x^4 in the binomial expansion of (x + 1)^5.
5
By the binomial theorem, the x^4 term has coefficient C(5, 1) × 1^1 = 5 × 1 = 5.
7. Find the coefficient of x^1 in the binomial expansion of (x + 5)^4.
500
By the binomial theorem, the x^1 term has coefficient C(4, 3) × 5^3 = 4 × 125 = 500.
8. Find the coefficient of x^5 in the binomial expansion of (x + 2)^6.
12
By the binomial theorem, the x^5 term has coefficient C(6, 1) × 2^1 = 6 × 2 = 12.
9. Find the coefficient of x^2 in the binomial expansion of (x + 7)^3.
21
By the binomial theorem, the x^2 term has coefficient C(3, 1) × 7^1 = 3 × 7 = 21.
10. Find the coefficient of x^2 in the binomial expansion of (x + 7)^4.
294
By the binomial theorem, the x^2 term has coefficient C(4, 2) × 7^2 = 6 × 49 = 294.
11. Find the coefficient of x^5 in the binomial expansion of (x + 5)^6.
30
By the binomial theorem, the x^5 term has coefficient C(6, 1) × 5^1 = 6 × 5 = 30.
12. Find the coefficient of x^2 in the binomial expansion of (x + 2)^5.
80
By the binomial theorem, the x^2 term has coefficient C(5, 3) × 2^3 = 10 × 8 = 80.

12 questions · Seed 43 · 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^4 in the binomial expansion of (x + 1)^6.
4 · 1 · 6 · Binomial theorem coefficie…
2. Find the coefficient of x^1 in the binomial expansion of (x + 3)^5.
1 · 3 · 5 · Binomial theorem coefficie…
3. Find the coefficient of x^1 in the binomial expansion of (x + 2)^3.
1 · 2 · 3 · Binomial theorem coefficie…
4. Find the coefficient of x^1 in the binomial expansion of (x + 1)^6.
1 · 1 · 6 · Binomial theorem coefficie…
5. Find the coefficient of x^4 in the binomial expansion of (x + 5)^6.
4 · 5 · 6 · Binomial theorem coefficie…
6. Find the coefficient of x^4 in the binomial expansion of (x + 1)^5.
4 · 1 · 5 · Binomial theorem coefficie…
7. Find the coefficient of x^1 in the binomial expansion of (x + 5)^4.
1 · 5 · 4 · Binomial theorem coefficie…
8. Find the coefficient of x^5 in the binomial expansion of (x + 2)^6.
5 · 2 · 6 · Binomial theorem coefficie…
9. Find the coefficient of x^2 in the binomial expansion of (x + 7)^3.
2 · 7 · 3 · Binomial theorem coefficie…
10. Find the coefficient of x^2 in the binomial expansion of (x + 7)^4.
2 · 7 · 4 · Binomial theorem coefficie…
11. Find the coefficient of x^5 in the binomial expansion of (x + 5)^6.
5 · 5 · 6 · Binomial theorem coefficie…
12. Find the coefficient of x^2 in the binomial expansion of (x + 2)^5.
2 · 2 · 5 · Binomial theorem coefficie…

12 questions · Seed 43 · 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^4 in the binomial expansion of (x + 1)^6.
15
By the binomial theorem, the x^4 term has coefficient C(6, 2) × 1^2 = 15 × 1 = 15.
2. Find the coefficient of x^1 in the binomial expansion of (x + 3)^5.
405
By the binomial theorem, the x^1 term has coefficient C(5, 4) × 3^4 = 5 × 81 = 405.
3. Find the coefficient of x^1 in the binomial expansion of (x + 2)^3.
12
By the binomial theorem, the x^1 term has coefficient C(3, 2) × 2^2 = 3 × 4 = 12.
4. Find the coefficient of x^1 in the binomial expansion of (x + 1)^6.
6
By the binomial theorem, the x^1 term has coefficient C(6, 5) × 1^5 = 6 × 1 = 6.
5. Find the coefficient of x^4 in the binomial expansion of (x + 5)^6.
375
By the binomial theorem, the x^4 term has coefficient C(6, 2) × 5^2 = 15 × 25 = 375.
6. Find the coefficient of x^4 in the binomial expansion of (x + 1)^5.
5
By the binomial theorem, the x^4 term has coefficient C(5, 1) × 1^1 = 5 × 1 = 5.
7. Find the coefficient of x^1 in the binomial expansion of (x + 5)^4.
500
By the binomial theorem, the x^1 term has coefficient C(4, 3) × 5^3 = 4 × 125 = 500.
8. Find the coefficient of x^5 in the binomial expansion of (x + 2)^6.
12
By the binomial theorem, the x^5 term has coefficient C(6, 1) × 2^1 = 6 × 2 = 12.
9. Find the coefficient of x^2 in the binomial expansion of (x + 7)^3.
21
By the binomial theorem, the x^2 term has coefficient C(3, 1) × 7^1 = 3 × 7 = 21.
10. Find the coefficient of x^2 in the binomial expansion of (x + 7)^4.
294
By the binomial theorem, the x^2 term has coefficient C(4, 2) × 7^2 = 6 × 49 = 294.
11. Find the coefficient of x^5 in the binomial expansion of (x + 5)^6.
30
By the binomial theorem, the x^5 term has coefficient C(6, 1) × 5^1 = 6 × 5 = 30.
12. Find the coefficient of x^2 in the binomial expansion of (x + 2)^5.
80
By the binomial theorem, the x^2 term has coefficient C(5, 3) × 2^3 = 10 × 8 = 80.

12 questions · Seed 43 · Age guidance, not curriculum certification