Binomial Theorem

We are asked to find the term independent of in the binomial expansion of .

The binomial expansion of is given by:

In this case, and , and we are expanding . The general term in the expansion is:

We can simplify this term:

For the term to be independent of , the exponent of must be 0. So, we set:

Solving for , we get:

Therefore, the term independent of corresponds to , and the term is:

The term independent of is .

We are given the expression and need to find the value of such that appears in the th term of its binomial expansion.

The binomial expansion of is given by:

In this case, and , and we are expanding . The general term in the expansion is:

We need to simplify this expression. The powers of in the terms are:

The power of in the general term is:

We are interested in the term where the power of is 22, so we set:

Solving for , we get:

The term corresponding to is the th term. Therefore, .

The value of is 4.

We are given the equation: Using Pascal's identity: we can apply this identity to simplify the left-hand side: Thus, we have the equation: Since the binomial coefficients are equal, we can equate the numbers: Solving for :

The correct option is (B) :

We are given that . To find the value of , substitute into the equation. This gives: Simplifying the left-hand side: Now simplify the right-hand side: Since this represents the sum of all coefficients, the value of the sum is simply , which is equivalent to . Therefore,

The correct option is (C) : 3¹⁰

In the binomial expansion of , we use the binomial theorem: Here, , , and . The general term in the expansion is: Simplifying: We need the term where the powers of and are 6 and 6, respectively. Thus, we set: This gives . Now, we calculate the corresponding term: The coefficient is:

The correct option is (A) :

Given: In the expansion of , the coefficients of the term and term are equal.

We know the general term in the binomial expansion is:

Therefore: 1. The term corresponds to 2. The term corresponds to

Setting their coefficients equal:

This equality holds when: 1. (which gives , invalid as must be positive integer), or 2. (by the symmetry property of binomial coefficients)

Solving the second case:

The correct answer is (E) 4.