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Binomial Theorem and Pascal's Triangle

A free Algebra II lesson from the “Sequences, Series, and Counting” unit, with a worked example and practice problems including step-by-step solutions.

Pascal's triangle is built so each entry is the sum of the two above it. Row n of Pascal's triangle gives the binomial coefficients of (a + b)^n. The kth entry (counting from 0) is C(n, k). For example, (a + b)^3 = 1 a^3 + 3 a^2 b + 3 a b^2 + 1 b^3 — coefficients 1, 3, 3, 1 from row 3.

What you'll learn

Why it matters: Probability calculations (Bernoulli trials), polynomial expansions in algebra and calculus, and combinatorics all use the binomial theorem.

Worked example

Problem. Find the coefficient of x^2 in (1 + x)^4.

  1. Row 4 of Pascal: 1, 4, 6, 4, 1.
  2. Coefficient of x^2 is the third entry (the C(4, 2) entry) = 6.

Answer: 6

Practice problems

1. Row 3 of Pascal: 1, _, _, 1. Enter the middle two entries separated by a comma (smallest first).

Show solution
  1. Warm-up: First identify exactly what the question is asking: Row 3 of Pascal: 1, _, _, 1. Enter the middle two entries separated by a comma (smallest first).
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Row 3: 1 3 3 1.
  4. Check the result by substituting or estimating: the response should match 3,3 and make sense in the original problem.

Answer: 3,3

2. Row 4 of Pascal: 1, 4, _, 4, 1. Enter the middle entry.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Row 4 of Pascal: 1, 4, _, 4, 1. Enter the middle entry.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Row 4: 1 4 6 4 1.
  4. Check the result by substituting or estimating: the response should match 6 and make sense in the original problem.

Answer: 6

3. Coefficient of x^2 in (1 + x)^3.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Coefficient of x^2 in (1 + x)^3.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Row 3 is 1 3 3 1; the x^2 entry is the third number = 3.
  4. Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.

Answer: 3

4. Coefficient of x^2 in (1 + x)^4.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Coefficient of x^2 in (1 + x)^4.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Row 4: 1 4 6 4 1; x^2 entry is 6.
  4. Check the result by substituting or estimating: the response should match 6 and make sense in the original problem.

Answer: 6

5. Coefficient of a^2 * b in (a + b)^3.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Coefficient of a^2 * b in (a + b)^3.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Row 3: 1 3 3 1; a^2 b entry is 3.
  4. Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.

Answer: 3

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