📘 BINOMIAL THEOREM NOTES
(Class 11/12 Standard - 24 Pages Approx)
Introduction to Binomial Theorem
A binomial is an algebraic expression with two terms, e.g., (a+b)(a + b)(a+b), (x−y)(x - y)(x−y),
etc.
The Binomial Theorem gives a formula for expanding powers of binomials like (a+b)n(a +
b)^n(a+b)n for any non-negative integer nnn.
Why It's Important:
● Essential in algebra and combinatorics.
● Frequently used in exams like JEE, NEET, and board exams.
Factorial Notation
Definition:
For a positive integer nnn,
n!=n×(n−1)×(n−2)×⋯×1n! = n \times (n-1) \times (n-2) \times \cdots \times
1n!=n×(n−1)×(n−2)×⋯×1
Examples:
● 4!=4×3×2×1=244! = 4 \times 3 \times 2 \times 1 = 244!=4×3×2×1=24
● 0!=10! = 10!=1 (By definition)
Combination Formula
Used to find the number of ways to choose rrr elements from a set of nnn elements.
, nCr=n!r!(n−r)!{}^nC_r = \frac{n!}{r!(n-r)!}nCr=r!(n−r)!n!
Properties:
● nC0=nCn=1{}^nC_0 = {}^nC_n = 1nC0=nCn=1
● nCr=nCn−r{}^nC_r = {}^nC_{n-r}nCr=nCn−r
Examples:
● 5C2=5!2!3!=10{}^5C_2 = \frac{5!}{2!3!} = 105C2=2!3!5!=10
Statement of Binomial Theorem
For any positive integer nnn:
(a+b)n=∑r=0nnCran−rbr(a + b)^n = \sum_{r=0}^n {}^nC_r a^{n-r} b^r(a+b)n=r=0∑nnCran−rbr
Each term in the expansion is of the form:
Tr+1=nCran−rbrT_{r+1} = {}^nC_r a^{n-r} b^rTr+1=nCran−rbr
Pascal’s Triangle
A triangular pattern of binomial coefficients:
markdown
CopyEdit
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
Each number is the sum of the two numbers directly above it.
(Class 11/12 Standard - 24 Pages Approx)
Introduction to Binomial Theorem
A binomial is an algebraic expression with two terms, e.g., (a+b)(a + b)(a+b), (x−y)(x - y)(x−y),
etc.
The Binomial Theorem gives a formula for expanding powers of binomials like (a+b)n(a +
b)^n(a+b)n for any non-negative integer nnn.
Why It's Important:
● Essential in algebra and combinatorics.
● Frequently used in exams like JEE, NEET, and board exams.
Factorial Notation
Definition:
For a positive integer nnn,
n!=n×(n−1)×(n−2)×⋯×1n! = n \times (n-1) \times (n-2) \times \cdots \times
1n!=n×(n−1)×(n−2)×⋯×1
Examples:
● 4!=4×3×2×1=244! = 4 \times 3 \times 2 \times 1 = 244!=4×3×2×1=24
● 0!=10! = 10!=1 (By definition)
Combination Formula
Used to find the number of ways to choose rrr elements from a set of nnn elements.
, nCr=n!r!(n−r)!{}^nC_r = \frac{n!}{r!(n-r)!}nCr=r!(n−r)!n!
Properties:
● nC0=nCn=1{}^nC_0 = {}^nC_n = 1nC0=nCn=1
● nCr=nCn−r{}^nC_r = {}^nC_{n-r}nCr=nCn−r
Examples:
● 5C2=5!2!3!=10{}^5C_2 = \frac{5!}{2!3!} = 105C2=2!3!5!=10
Statement of Binomial Theorem
For any positive integer nnn:
(a+b)n=∑r=0nnCran−rbr(a + b)^n = \sum_{r=0}^n {}^nC_r a^{n-r} b^r(a+b)n=r=0∑nnCran−rbr
Each term in the expansion is of the form:
Tr+1=nCran−rbrT_{r+1} = {}^nC_r a^{n-r} b^rTr+1=nCran−rbr
Pascal’s Triangle
A triangular pattern of binomial coefficients:
markdown
CopyEdit
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
Each number is the sum of the two numbers directly above it.
