Quick Revision Notes: Class 11 Mathematics

Chapter: Binomial Theorem (द्विपद प्रमेय)

---

### Concept 1: Introduction to Binomial Theorem

A binomial expression is an algebraic expression consisting of two terms, connected by '+' or '-' sign. For example: (x + y), (a - b)^2, etc.

For any positive integer $n$, the Binomial Theorem is given by: (x + y)^n = C(n,0) x^n y^0 + C(n,1) x^{n-1} y^1 + C(n,2) x^{n-2} y^2 + ... + C(n,r) x^{n-r} y^r + ... + C(n,n) x^0 y^n

Using summation notation: (x + y)^n = sum_{r=0}^{n} C(n,r) x^{n-r} y^r

Where:

• $C(n,r)$ or ^nC_r denotes combinations and is equal to n! / [r! (n-r)!]
• The number of terms in the expansion of (x + y)^n is n + 1.

---

### Key Formulas & Properties

1. Factorial Notation (n!): n! = n × (n-1) × (n-2) × ... × 3 × 2 × 1 Note: 0! = 1 and 1! = 1

2. Properties of Binomial Coefficients (^nC_r):

   • ^nC_0 = ^nC_n = 1
   • ^nC_1 = ^nC_{n-1} = n
   • Complementary Property: ^nC_r = ^nC_{n-r}
   • Pascal's Rule: ^nC_r + ^nC_{r-1} = ^{n+1}C_r
   • Sum of all binomial coefficients: ^nC_0 + ^nC_1 + ^nC_2 + ... + ^nC_n = 2^n
   • Sum of odd binomial coefficients = Sum of even binomial coefficients: ^nC_0 + ^nC_2 + ^nC_4 + ... = ^nC_1 + ^nC_3 + ^nC_5 + ... = 2^{n-1}

---

### Concept 2: General Term (व्यापक पद)

The general term in the expansion of (x + y)^n is denoted by T_{r+1} and is given by: T_{r+1} = ^nC_r x^{n-r} y^r

• Important Points:
  • To find the coefficient of x^k, equate the index of x in the general term to k, find r, and substitute it back.
  • Terms are numbered starting from T_1 for r = 0.

---

### Concept 3: Middle Term(s) (मध्य पद)

The middle term(s) depend upon the value of $n$:

1. When $n$ is Even (सम):

   • There is only one middle term.
   • Middle term = T_{(n/2) + 1}

2. When $n$ is Odd (विषम):

   • There are two middle terms.
   • Middle terms = T_{(n+1)/2} and T_{[(n+1)/2] + 1}

---

### Concept 4: Independent Term / Constant Term (अचर पद / स्वतंत्र पद)

• The term independent of $x$ (or constant term) in an expansion is that term in the general term where the total power of $x$ equals zero.
• Put the power of $x$ equal to 0 to find $r$, then substitute $r$ to get the value of the term.

---

### Concept 5: Properties of (1 + x)^n

By substituting x = 1 and y = x in the general binomial theorem, we get: (1 + x)^n = ^nC_0 + ^nC_1 x + ^nC_2 x^2 + ... + ^nC_n x^n

• Some useful expansions:
  • (1 - x)^n = ^nC_0 - ^nC_1 x + ^nC_2 x^2 - ^nC_3 x^3 + ... + (-1)^n ^nC_n x^n
  • (1 + x)^n + (1 - x)^n = 2 [^nC_0 + ^nC_2 x^2 + ^nC_4 x^4 + ...]
  • (1 + x)^n - (1 - x)^n = 2 [^nC_1 x + ^nC_3 x^3 + ^nC_5 x^5 + ...]

---

### MP Board Examination Tips (परीक्षा के लिए विशेष टिप्स)

1. Formula Memorization: Always write the formula T_{r+1} = ^nC_r x^{n-r} y^r before solving questions related to the general term to ensure you get step-marking.
2. Sign Caution: Be extremely careful with negative signs in expressions like (x - y)^n. Substitute y along with its negative sign into the formula.
3. Factorial Calculations: Practice basic factorial calculations to save time during the exam.