Binomial theorem class 11

 𝗕𝗶𝗻𝗼𝗺𝗶𝗮𝗹 𝗧𝗵𝗲𝗼𝗿𝗲𝗺

1. Introduction to Binomial Theorem

The binomial theorem is a fundamental concept in algebra, which allows us to expand expressions of the form (a + b)^n, where "a" and "b" are constants and "n" is a positive integer.

2. Binomial Theorem for Positive Integral Exponents

Binomial Theorem Formula: For any positive integer "n," the expansion of (a + b)^n is given by:

(a + b)^n = C(n, 0) * a^n * b^0 + C(n, 1) * a^(n-1) * b^1 + C(n, 2) * a^(n-2) * b^2 + ... + C(n, n) * a^0 * b^n

Where:

"C(n, k)" represents the binomial coefficient, calculated as C(n, k) = n! / (k!(n-k)!).

"a^n" represents "a" raised to the power of "n."

"b^k" represents "b" raised to the power of "k."

3. Middle Term(s) in Binomial Expansion

To find the middle term(s) in the binomial expansion, we use the formula for the middle term:

Middle term(s) = (n + 1) / 2 if "n" is odd.

Middle terms = [n/2]th term and [(n/2) + 1]th term if "n" is even.

4. General Term in Binomial Expansion

The formula for the "r"-th term in the binomial expansion is:

Term(r) = C(n, r) * a^(n-r) * b^r

5. Binomial Theorem for Any Rational Exponent

The binomial theorem can be extended to any rational exponent. If "n" is not a positive integer, the expansion of (a + b)^n is done using the formula:

(a + b)^n = ∑ [r=0 to ∞] (n choose r) * a^(n-r) * b^r

6. Binomial Coefficients and Properties

Properties of binomial coefficients include Pascal's triangle.

Pascal's Triangle: Each number is the sum of the two numbers directly above it.

7. Applications of Binomial Theorem

Combinatorial problems.

Finding coefficients in polynomial expansions.

Probability problems.

8. JEE Main Questions with Answers

Here are some example JEE Main questions on the binomial theorem:

Question 1: Expand (x + y)^5.

Answer 1:

Using the binomial theorem formula:

(a + b)^n = C(n, 0) * a^n * b^0 + C(n, 1) * a^(n-1) * b^1 + ... + C(n, n) * a^0 * b^n

In this case, a = x, b = y, and n = 5.

(x + y)^5 = C(5, 0) * x^5 * y^0 + C(5, 1) * x^4 * y^1 + C(5, 2) * x^3 * y^2 + C(5, 3) * x^2 * y^3 + C(5, 4) * x^1 * y^4 + C(5, 5) * x^0 * y^5

Now, calculate the binomial coefficients:

C(5, 0) = 1

C(5, 1) = 5

C(5, 2) = 10

C(5, 3) = 10

C(5, 4) = 5

C(5, 5) = 1

So, the expansion is:

(x + y)^5 = x^5 + 5x^4y + 10x^3y^2 + 10x^2y^3 + 5xy^4 + y^5

Question 2: Find the coefficient of x^3 in the expansion of (1 + 2x)^5.

Answer 2:

We need to find the coefficient of x^3 in the expansion of (1 + 2x)^5.

Using the binomial theorem formula for (a + b)^n:

(a + b)^n = C(n, 0) * a^n * b^0 + C(n, 1) * a^(n-1) * b^1 + ... + C(n, n) * a^0 * b^n

In this case, a = 1, b = 2x, and n = 5.

(1 + 2x)^5 = C(5, 0) * 1^5 * (2x)^0 + C(5, 1) * 1^4 * (2x)^1 + C(5, 2) * 1^3 * (2x)^2 + C(5, 3) * 1^2 * (2x)^3 + C(5, 4) * 1^1 * (2x)^4 + C(5, 5) * 1^0 * (2x)^5

Now, calculate the binomial coefficients:

C(5, 0) = 1

C(5, 1) = 5

C(5, 2) = 10

C(5, 3) = 10

C(5, 4) = 5

C(5, 5) = 1

So, the expansion is:

(1 + 2x)^5 = 1 + 10x + 40x^2 + 80x^3 + 80x^4 + 32x^5

The coefficient of x^3 is 80.