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BINOMIAL THEOREM (POSITIVE INTEGER n)
(a+b)^n = sum_{r=0}^{n} nCr a^{n-r} b^r . There are n+1 terms. Coefficients are a row of Pascal's triangle. Meaning: you expand by choosing b from r of the n brackets (the rest contribute a). a and b can be numbers or letters, including (1+x)^n and (2x-3y)^n.
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BINOMIAL THEOREM (POSITIVE INTEGER n)
What is it?
(a+b)^n = sum_{r=0}^{n} nCr a^{n-r} b^r . There are n+1 terms. Coefficients are a row of Pascal's triangle. Meaning: you expand by choosing b from r of the n brackets (the rest contribute a). a and b can be numbers or letters, including (1+x)^n and (2x-3y)^n.
In simple words
- (a+b)n = sumr=0n nCr an-r br .
- There are n+1 terms.
- Coefficients are a row of Pascal's triangle.
- The power of a plus the power of b is always n. Use that to find r.
Important terms
BINOMIAL THEOREM (POSITIVE INTEGER n): (a+b)^n = sum_{r=0}^{n} nCr a^{n-r} b^r . There are n+1 terms. Coefficients are a row of Pascal's triangle. Meaning: you expand by choosing b from r of the n brackets (the rest contribute a). a and b can be numbers or letters, including (1+x)^n and (2x-3y)^n.
1. THE SUM: r runs 0 to n. First term an, last term bn. General term Tr+1 = nCr an-r br (the (r+1)th term, r starting at 0).
2. SMALL CHECK: (a+b)^2 = a^2+2ab+b^2. (a+b)^3 = a^3+3a^2 b+3 a b^2+b^3. Pascal: 1,4,6,4,1 for n=4.
What you should learn
- State the main ideas and notation used in binomial theorem (positive integer n).
- Apply the rules of binomial theorem (positive integer n) in a clear, step-by-step solution.
- Check signs, conditions, and the final answer for mathematical accuracy.
Examples
- T_{r+1} general
- nC0=1 first
- nCn=1 last
THE SUM
r runs 0 to n. First term an, last term bn. General term Tr+1 = nCr an-r br (the (r+1)th term, r starting at 0).
SMALL CHECK
(a+b)2 = a2+2ab+b2. (a+b)3 = a3+3a2 b+3 a b2+b3. Pascal: 1,4,6,4,1 for n=4.
Recommended for Class 11
Full school-level explanation
(a+b)^n = sum_{r=0}^{n} nCr a^{n-r} b^r . There are n+1 terms. Coefficients are a row of Pascal's triangle. Meaning: you expand by choosing b from r of the n brackets (the rest contribute a). a and b can be numbers or letters, including (1+x)^n and (2x-3y)^n.
Write the sum, then a small n:
WHAT — BINOMIAL THEOREM (POSITIVE INTEGER n)
(a+b)^n = sum_{r=0}^{n} nCr a^{n-r} b^r . There are n+1 terms. Coefficients are a row of Pascal's triangle. Meaning: you expand by choosing b from r of the n brackets (the rest contribute a). a and b can be numbers or letters, including (1+x)^n and (2x-3y)^n.
RULE — 1. THE SUM
r runs 0 to n. First term an, last term bn. General term Tr+1 = nCr an-r br (the (r+1)th term, r starting at 0).
METHOD — 2. SMALL CHECK
(a+b)2 = a2+2ab+b2. (a+b)3 = a3+3a2 b+3 a b2+b3. Pascal: 1,4,6,4,1 for n=4.
WORKED EXAMPLE
Worked: (x+2)4, term in x2. a=x, b=2, want x4-r 2r with 4-r=2 so r=2 T = 4C2 x2 22 = 6 x2 * 4 = 24 x2
THE SUM
r runs 0 to n. First term an, last term bn. General term Tr+1 = nCr an-r br (the (r+1)th term, r starting at 0).
Remember
- r = 0 to n
Examples
- T_{r+1} general
- nC0=1 first
- nCn=1 last
SMALL CHECK
(a+b)2 = a2+2ab+b2. (a+b)3 = a3+3a2 b+3 a b2+b3. Pascal: 1,4,6,4,1 for n=4.
Remember
- Pascal row n
Examples
- n=3: 1 3 3 1
- Signs if (a-b): (-1)^r
- (a-b)n = sum nCr an-r (-b)r
How to answer a binomial theorem (positive integer n) question
How it works
- Identify the given quantities, notation, and required result.
- Choose the matching rule and write it before substitution.
- Work one algebraic step at a time, keeping brackets and signs clear.
- Check restrictions, simplify the result, and verify it where possible.
Remember
- n+1 terms
- coeff of an-r br is nCr
- 24 x^2
- powers add to n
- r = 0 to n
- Pascal row n
- n+1 terms
- coeff of an-r br is nCr
How the ideas connect
Worked: (x+2)4, term in x2. a=x, b=2, want x4-r 2r with 4-r=2 so r=2 T = 4C2 x2 22 = 6 x2 * 4 = 24 x2
Deeper understanding
Go beyond memorising BINOMIAL THEOREM (POSITIVE INTEGER n). Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Worked: (x+2)4, term in x2. a=x, b=2, want x4-r 2r with 4-r=2 so r=2 T = 4C2 x2 22 = 6 x2 * 4 = 24 x2
Common mistakes
Exam focus
Core Idea
1. THE SUM
2. SMALL CHECK
Worked Example
Exam focus
Quick revision
- n+1 terms
- coeff of an-r br is nCr
- 24 x^2
- powers add to n
- BINOMIAL THEOREM (POSITIVE INTEGER n): (a+b)n = sumr=0n nCr an-r br . There are n+1 terms. Coefficients are a row of Pascal's tr
- 1. THE SUM: r runs 0 to n. First term an, last term bn. General term Tr+1 = nCr an-r br (the (r+1)th te
The power of a plus the power of b is always n. Use that to find r.
For long answers on BINOMIAL THEOREM (POSITIVE INTEGER n): define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- n+1 terms
- coeff of an-r br is nCr
- 24 x^2
- powers add to n
Important formulas
- n+1 terms
- coeff of an-r br is nCr
- 24 x^2
Common exam traps
- The power of a plus the power of b is always n. Use that to find r.
- Do not use a remembered formula before checking that its conditions fit the question.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
The power of a plus the power of b is always n. Use that to find r.
- Mistake
Do not use a remembered formula before checking that its conditions fit the question.
- Mistake
Do not hide sign changes or restrictions; write them on the line where they are used.
- Mistake
The power of a plus the power of b is always n. Use that to find r.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
n+1 terms
coeff of an-r br is nCr
24 x^2
powers add to n
(a+b)n = sumr=0n nCr an-r br . There are n+1 terms. Coefficients are a row of Pascal's triangle. Meaning: you expand by choosing b from r of the n br
r = 0 to n
- Definition
- (a+b)n = sumr=0n nCr an-r br . There are n+1 terms. Coefficients are a row of Pascal's triangle. Meaning: you expand by choosing b from r of the n br
- Formula
- r = 0 to n
- Example
- T_{r+1} general
- Critical warning
- The power of a plus the power of b is always n. Use that to find r.
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Exam Practice
Define
Q1. Define BINOMIAL THEOREM (POSITIVE INTEGER n).
Explain
Q2. Explain: THE SUM.
Example
Q3. Give one clear example of BINOMIAL THEOREM (POSITIVE INTEGER n) and say why it fits.
Mistake
Q4. State one common mistake students make about BINOMIAL THEOREM (POSITIVE INTEGER n).
Exam note
Q5. Write a short exam note on BINOMIAL THEOREM (POSITIVE INTEGER n): definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.