Binomial Theorem

Step 1: Total binary sequences of length .
Each position has 2 choices (0 or 1), so:

Step 2: Use symmetry (even-odd parity).
In all binary sequences, number of sequences with even number of 0's equals number of sequences with odd number of 0's.
Because flipping the first bit maps even-zero sequences to odd-zero sequences one-to-one.
Step 3: Hence count of even-zero sequences.

Final Answer:

Step 1: Write general term.
In expansion of , general term is:
Step 2: Substitute .
So term becomes:
Numerical value is:
Step 3: Find greatest term using ratio.
We need greatest term where ratio just becomes less than 1.
Check for :
So terms increasing till .
Thus greatest term is at , i.e. .
Step 4: Compute .
Final Answer:

we have



Now,



Therefore, and terms are numerically greatest in the expansion of .
Hence, greatest term =



and

Hence Greatest term (numerically)

Step 1: Understand the expansion.
The sum of the coefficients of any binomial expansion can be found by substituting and in the expansion.
Step 2: Conclusion.
Thus, the sum of the coefficients is .
Final Answer:

For the sum of the coefficients in the expansion of , put
Sum of the coefficients

Step 1: Understanding the given expansion.
We are given the summation: This is the binomial expansion of , or equivalently .
Step 2: Expanding the expression.
Using the binomial theorem, we expand : Thus, the general term in the expansion is: To find the coefficient of , we need the term where the exponent of is 53. This occurs when:
Step 3: Finding the coefficient.
When , the corresponding term is: The coefficient of is therefore: Since the problem asks for the coefficient of , the answer is , and the correct answer is (c).

We are asked to find the coefficient of in the expansion of . Step 1: Using the expansion of . We know that the expansion of for is given by: This is the standard Taylor series expansion for . Step 2: Applying the change of base formula. We are given , which can be written in terms of the natural logarithm as: Substitute the expansion of into this: Step 3: Finding the coefficient of . The coefficient of in the expansion of is: Since , we have: Thus, the coefficient of in the expansion of is .
Step 4: Conclusion.
Therefore, the correct answer is (b).

Step 1: Understanding the series.
The given series is a binomial expansion series that can be simplified using the general formula for a geometric series. After evaluating, the sum is found to be .

Step 2: Conclusion.
The correct expansion value is , corresponding to option (3).

Step 1: Identifying the expansion.
The expansion of will result in terms of the form . The coefficient of is found by using the general form of the expansion and applying the binomial theorem.

Step 2: Conclusion.
The coefficient of is , corresponding to option (3).

Using the binomial expansion formula, we compare the terms and solve for . This gives us the value of as 12.
Final Answer: