Question

The greatest term in expansion of \((3+2 \mathrm{x})^{50}\) is \(;\) where \(\mathrm{x}=(1 / 5)\) (a) \({ }^{50} \mathrm{C}_{7} 343(2 / 5)^{7}\) (b) \({ }^{50} \mathrm{C}_{6} 3^{44}(2 / 5)^{6}\) (c) \({ }^{50} \mathrm{C}_{43} 3^{7}(2 / 5)^{43}\) (d) \({ }^{50} \mathrm{C}_{44} 3^{6}(2 / 5)^{44}\)

Short answer

(b) \({ }^{50}C_{6} 3^{44}(2 / 5)^{6}\)

Step-by-step solution

Write down the general term formula for binomial expansion

The general formula for the term in a binomial expansion \((a+b)^{n}\) is: \(t_r = ^{n}C_{r-1}a^{n-r+1}b^{r-1}\), where \(r\) denotes the position (order) of the term in the expansion.

Convert each given term into its general term formula

We will convert the given terms in the options (a, b, c, and d) into their general term formula format. (a) \({ }^{50}C_{7} 343(2 / 5)^{7}\) can be rewritten as \({ }^{50}C_{6} 3^{44}(2 / 5)^{6}\) (b) \({ }^{50}C_{6} 3^{44}(2 / 5)^{6}\) is already in its general term format (c) \({ }^{50}C_{43} 3^{7}(2 / 5)^{43}\) can be rewritten as \({ }^{50}C_{42} 3^{8}(2 / 5)^{42}\) (d) \({ }^{50}C_{44} 3^{6}(2 / 5)^{44}\) can be rewritten as \({ }^{50}C_{43} 3^{7}(2 / 5)^{43}\)

Compare the coefficients to find the greatest term

Now, let's write down the coefficients of these terms: 1. Coefficient of \({ }^{50}C_{6} 3^{44}(2 / 5)^{6} = ^{50}C_{6} 3^{44}\) 2. Coefficient of \({ }^{50}C_{42} 3^{8}(2 / 5)^{42} = ^{50}C_{42} 3^{8}\) 3. Coefficient of \({ }^{50}C_{43} 3^{7}(2 / 5)^{43} = ^{50}C_{43} 3^{7}\) Based on binomial theorem property, the coefficients of the terms will be maximum at the center and minimum at the ends. In the given binomial, \(n=50\) is an even number, so the two central terms will have the maximum coefficients; these are terms with \(r=26\) and \(r=25\). From the given options, \({ }^{50}C_{6} 3^{44}(2 / 5)^{6}\) has the greatest index powers of \(3\) in comparison to the others. Therefore, the greatest term in the expansion of \((3+2x)^{50}\) is:

Answer

(b) \({ }^{50}C_{6} 3^{44}(2 / 5)^{6}\)

Key concepts

Binomial Expansion

The binomial expansion is the process of expanding an expression that is raised to a power and consists of two terms. This expansion follows a specific pattern known as Pascal's Triangle, and each term of the expansion is represented by the binomial coefficients. Let's take a closer look at how a binomial expression like \( (a+b)^n \) is expanded.

Using the binomial theorem, this expression expands to the sum of terms that include \( a \) raised to a decreasing power and \( b \) raised to an increasing power, with binomial coefficients pre-multiplying each term. The general term in this expansion is given as \( t_r = ^nC_{r-1} a^{n-r+1} b^{r-1} \), where \( r \) is the term number.

For example, in the exercise provided, the expansion of \( (3+2x)^{50} \) involves 51 terms, where \( x = 1/5 \) and the greatest term is identified by finding the maximum value amongst the expanded terms following the binomial theorem.

Combinatorics

Combinatorics is a branch of mathematics dealing with combinations, arrangements, and counting. Within the binomial expansion, combinatorics is essential because the binomial coefficients represent the number of ways to choose \( r-1 \) elements from a set of \( n \) elements, without considering the order of selection. This number is also denoted as \( ^nC_{r-1} \), which is read as 'n choose r-1'.

These coefficients are calculated using the formula:\[ ^nC_{r-1} = \frac{n!}{(n-r+1)!(r-1)!} \], where \( ! \) denotes the factorial of a number. This usage of combinatorics ensures that each term in the binomial expansion has the correct coefficient that corresponds to the number of ways that term would be generated via the distributive property if the binomial were expanded manually.

Binomial Coefficients

Binomial coefficients are the numerical factors that multiply the terms in the expansion of a binomial expression. They are central to understanding the binomial theorem and are found in Pascal's Triangle. These coefficients are represented by \( ^nC_r \) in the expansion of \( (a+b)^n \).

The binomial coefficients are symmetric, meaning that \( ^nC_r = ^nC_{n-r} \), which reflects the property of the coefficients being mirrored across the middle of Pascal's Triangle. In the context of the given problem, for the binomial expression \( (3+2x)^{50} \) and \( x = 1/5 \), we calculate the appropriate binomial coefficients to determine the greatest term.

The given options present terms with different binomial coefficients, and to find the greatest term, we examine these coefficients alongside the powers of 3 and \( (2/5) \) for each term. The coefficients play a critical role in identifying the term with the largest value in the expansion.