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

12 questions · Seed 104730 · 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^2 in the binomial expansion of (x + 1)^6.
15
By the binomial theorem, the x^2 term has coefficient C(6, 4) × 1^4 = 15 × 1 = 15.
2. Find the coefficient of x^2 in the binomial expansion of (x + 7)^6.
36015
By the binomial theorem, the x^2 term has coefficient C(6, 4) × 7^4 = 15 × 2401 = 36015.
3. 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.
4. 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.
5. Find the coefficient of x^2 in the binomial expansion of (x + 5)^5.
1250
By the binomial theorem, the x^2 term has coefficient C(5, 3) × 5^3 = 10 × 125 = 1250.
6. Find the coefficient of x^2 in the binomial expansion of (x + 1)^4.
6
By the binomial theorem, the x^2 term has coefficient C(4, 2) × 1^2 = 6 × 1 = 6.
7. Find the coefficient of x^4 in the binomial expansion of (x + 4)^5.
20
By the binomial theorem, the x^4 term has coefficient C(5, 1) × 4^1 = 5 × 4 = 20.
8. Find the coefficient of x^2 in the binomial expansion of (x + 3)^5.
270
By the binomial theorem, the x^2 term has coefficient C(5, 3) × 3^3 = 10 × 27 = 270.
9. Find the coefficient of x^2 in the binomial expansion of (x + 6)^4.
216
By the binomial theorem, the x^2 term has coefficient C(4, 2) × 6^2 = 6 × 36 = 216.
10. Find the coefficient of x^3 in the binomial expansion of (x + 6)^4.
24
By the binomial theorem, the x^3 term has coefficient C(4, 1) × 6^1 = 4 × 6 = 24.
11. Find the coefficient of x^3 in the binomial expansion of (x + 4)^4.
16
By the binomial theorem, the x^3 term has coefficient C(4, 1) × 4^1 = 4 × 4 = 16.
12. Find the coefficient of x^1 in the binomial expansion of (x + 5)^5.
3125
By the binomial theorem, the x^1 term has coefficient C(5, 4) × 5^4 = 5 × 625 = 3125.

12 questions · Seed 104730 · 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^2 in the binomial expansion of (x + 1)^6.
2 · 1 · 6 · Binomial theorem coefficie…
2. Find the coefficient of x^2 in the binomial expansion of (x + 7)^6.
2 · 7 · 6 · Binomial theorem coefficie…
3. Find the coefficient of x^5 in the binomial expansion of (x + 5)^6.
5 · 5 · 6 · Binomial theorem coefficie…
4. Find the coefficient of x^2 in the binomial expansion of (x + 2)^6.
2 · 2 · 6 · Binomial theorem coefficie…
5. Find the coefficient of x^2 in the binomial expansion of (x + 5)^5.
2 · 5 · 5 · Binomial theorem coefficie…
6. Find the coefficient of x^2 in the binomial expansion of (x + 1)^4.
2 · 1 · 4 · Binomial theorem coefficie…
7. Find the coefficient of x^4 in the binomial expansion of (x + 4)^5.
4 · 4 · 5 · Binomial theorem coefficie…
8. Find the coefficient of x^2 in the binomial expansion of (x + 3)^5.
2 · 3 · 5 · Binomial theorem coefficie…
9. Find the coefficient of x^2 in the binomial expansion of (x + 6)^4.
2 · 6 · 4 · Binomial theorem coefficie…
10. Find the coefficient of x^3 in the binomial expansion of (x + 6)^4.
3 · 6 · 4 · Binomial theorem coefficie…
11. Find the coefficient of x^3 in the binomial expansion of (x + 4)^4.
3 · 4 · 4 · Binomial theorem coefficie…
12. Find the coefficient of x^1 in the binomial expansion of (x + 5)^5.
1 · 5 · 5 · Binomial theorem coefficie…

12 questions · Seed 104730 · 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^2 in the binomial expansion of (x + 1)^6.
15
By the binomial theorem, the x^2 term has coefficient C(6, 4) × 1^4 = 15 × 1 = 15.
2. Find the coefficient of x^2 in the binomial expansion of (x + 7)^6.
36015
By the binomial theorem, the x^2 term has coefficient C(6, 4) × 7^4 = 15 × 2401 = 36015.
3. 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.
4. 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.
5. Find the coefficient of x^2 in the binomial expansion of (x + 5)^5.
1250
By the binomial theorem, the x^2 term has coefficient C(5, 3) × 5^3 = 10 × 125 = 1250.
6. Find the coefficient of x^2 in the binomial expansion of (x + 1)^4.
6
By the binomial theorem, the x^2 term has coefficient C(4, 2) × 1^2 = 6 × 1 = 6.
7. Find the coefficient of x^4 in the binomial expansion of (x + 4)^5.
20
By the binomial theorem, the x^4 term has coefficient C(5, 1) × 4^1 = 5 × 4 = 20.
8. Find the coefficient of x^2 in the binomial expansion of (x + 3)^5.
270
By the binomial theorem, the x^2 term has coefficient C(5, 3) × 3^3 = 10 × 27 = 270.
9. Find the coefficient of x^2 in the binomial expansion of (x + 6)^4.
216
By the binomial theorem, the x^2 term has coefficient C(4, 2) × 6^2 = 6 × 36 = 216.
10. Find the coefficient of x^3 in the binomial expansion of (x + 6)^4.
24
By the binomial theorem, the x^3 term has coefficient C(4, 1) × 6^1 = 4 × 6 = 24.
11. Find the coefficient of x^3 in the binomial expansion of (x + 4)^4.
16
By the binomial theorem, the x^3 term has coefficient C(4, 1) × 4^1 = 4 × 4 = 16.
12. Find the coefficient of x^1 in the binomial expansion of (x + 5)^5.
3125
By the binomial theorem, the x^1 term has coefficient C(5, 4) × 5^4 = 5 × 625 = 3125.

12 questions · Seed 104730 · Age guidance, not curriculum certification