Problem 18 - A Complete Resource Book in Mathematics for JEE Main

Question

The sum of the last ten coefficients in the expansion of \((1+x)^{19}\) when expanded in ascending powers of \(x\) is (A) \(2^{18}\) (B) \(2^{19}\) (C) \(2^{18}-{ }^{19} C_{10}\) (D) \(\frac{1}{2}\left(2^{19}-1\right)\)

Step-by-Step Solution

Answer

The sum of the last ten coefficients is \(2^{18}\).

Step 1: Understanding the Problem

We need to determine the sum of the last ten coefficients in the expansion of \((1+x)^{19}\). The coefficients are given by the binomial expansion formula \(^nC_k\) from \(k=0\) to \(k=19\). We need to find the sum of the coefficients from \( ^{19}C_{10} \) to \( ^{19}C_{19} \).

Step 2: Sum of All Coefficients

The sum of all the coefficients in the expansion of \((1+x)^{n}\) is given by substituting \(x=1\) into the expression, which yields \(2^n\). Therefore, for \((1+x)^{19}\), the sum of all the coefficients is \(2^{19}\).

Step 3: Sum of First Ten Coefficients

Similarly, the sum of the first ten coefficients \(\binom{19}{0}\), \(\binom{19}{1}\), ..., \(\binom{19}{9}\) can be found because the symmetry of the binomial coefficients around the middle term allows us to use the same principle as above when suitably adjusted, differing by central terms.

Step 4: Using Symmetry

Notice the symmetry in the binomial coefficients: \(\binom{19}{k} = \binom{19}{19-k}\). This means the sum of the first ten coefficients is the same as the sum of the last ten coefficients. The first nine, plus half of the middle (19th) coefficient gives half the total: \(2^{18}\). Hence the sum of the last ten is the other half, \(2^18\), with a total symmetry.

Step 5: Calculating Desired Sum

Thus, the sum of the last ten coefficients is:\[ \frac{1}{2}(2^{19}) = 2^{18}\].

Step 6: Verify the Result

The calculation above shows that due to the symmetry properties of binomial expansions, each half sums equally to \(2^{18}\). Thus, option (A) is correct, \(2^{18}\).

Key Concepts

Binomial Expansion

The binomial theorem provides a systematic way to expand expressions of the form \( (a+b)^n \). This expansion involves terms of the form \( ^nC_k a^{n-k} b^k \), where \( n \) is a positive integer, \( k \) is a particular term's position, and \( ^nC_k \) is the binomial coefficient that determines the weight of each term.
The expression is expanded as follows:

• Start with \( a^n \) and end with \( b^n \).
• Each step involves reducing the power of \( a \) by one and increasing the power of \( b \) by one.
• The binomial coefficients \( ^nC_k \) can be determined using Pascal's triangle or the formula \( ^nC_k = \frac{n!}{k!(n-k)!} \).

In the context of our problem, \( (1+x)^{19} \) is expanded as \( \sum_{k=0}^{19} \binom{19}{k} x^k \). Each term is associated with a binomial coefficient \( \binom{19}{k} \), and understanding this structure is key to analyzing the expansion.

Symmetry in Coefficients

One of the most intriguing aspects of binomial expansions is the symmetry of the coefficients. The coefficients in the expansion \( (a+b)^n \) have a symmetric property, meaning that \( \binom{n}{k} = \binom{n}{n-k} \).
This symmetry arises because each coefficient counts the same number of ways to choose \( k \) items from \( n \) as choosing \( n-k \) items from \( n \), essentially just rearranging the selection.

• This is beautifully demonstrated in Pascal's triangle, where each row reads the same forward and backward.
• For example, in our case, \( \binom{19}{10} = \binom{19}{9} \).

The important implication for our exercise is that the sum of coefficients from the beginning of the expansion matches the sum of coefficients at the end.
This property allows us to simplify computations significantly, particularly when determining sums of specific parts of the expansion.

Binomial Coefficient Properties

Binomial coefficients \( ^nC_k \) have several interesting and useful properties that play a crucial role in the application of the binomial theorem.
One key feature is that the sum of the coefficients for a specific binomial expansion \( (a+b)^n \) when both variables are set to one (\