Problem 52 - A Complete Resource Book in Mathematics for JEE Main

Question

The sum of the series \(\sum_{r=0}^{10}{\underline{\phantom{xx}}}^{20} C_{r}\) is (A) \(2^{19}-\frac{1}{2} \cdot{ }^{20} C_{10}\) (B) \(2^{19}+\frac{1}{2} \cdot{ }^{20} C_{10}\) (C) \(2^{19}\) (D) \(2^{20}\)

Step-by-Step Solution

Answer

The sum of the series is \(2^{19}\), which is option (C).

Step 1: Understand the Series

The given series is \( \sum_{r=0}^{10} \binom{20}{r} \). This means we are summing the binomial coefficients \( \binom{20}{0}, \binom{20}{1}, \ldots, \binom{20}{10} \).

Step 2: Use the Binomial Theorem

According to the binomial theorem, the sum of all binomial coefficients for a given number \( n \) is \( 2^n \). Thus, \( \sum_{r=0}^{20} \binom{20}{r} = 2^{20} \).

Step 3: Compute the Complement Fraction

Notice that \( \sum_{r=0}^{10} \binom{20}{r} \) represents only half the terms of the full binomial expansion. This is symmetric such that \( \sum_{r=0}^{10} \binom{20}{r} = \sum_{r=11}^{20} \binom{20}{r} \). Thus, \( 2 \sum_{r=0}^{10} \binom{20}{r} = 2^{20} \).

Step 4: Solve for the Desired Sum

From Step 3, \( \sum_{r=0}^{10} \binom{20}{r} = \frac{2^{20}}{2} = 2^{19} \). It matches option (C).

Key Concepts

Binomial Coefficients

Binomial coefficients are essential components in the field of combinatorics and are heavily used in the binomial theorem for expansions. A binomial coefficient is represented as \( \binom{n}{r} \), which is read as "n choose r". It tells us the number of ways to choose \( r \) elements from a set of \( n \) elements. The formula to calculate a binomial coefficient is:\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]Here, \( n! \) denotes the factorial of \( n \), which is the product of all positive integers up to \( n \). Binomial coefficients are symmetrical, meaning \( \binom{n}{r} = \binom{n}{n-r} \), which plays a key role in simplifying calculations within the binomial expansion.
Binomial coefficients appear in the expansion of a binomial expression raised to a power, such as \((a+b)^n \), where they form Pascal's triangle, a triangular array of the coefficients calculated by a recursive method. Understanding binomial coefficients allows one to find specific terms in a polynomial expansion without fully expanding it.

Symmetric Expansion

Symmetric expansion relies on the property of binomial coefficients being symmetrical. This symmetry is generally expressed as \( \binom{n}{r} = \binom{n}{n-r} \). In a full binomial expansion like \((a+b)^n\), each term is mirrored around the midpoint of the expansion.
For instance, in the binomial expansion of \((a + b)^{20} \), the terms from \( \binom{20}{0} \) to \( \binom{20}{10} \) mirror those from \( \binom{20}{20} \) to \( \binom{20}{11} \). This means that both halves of the expansion are identical when summed. If one calculates only half the terms, they can determine the sum of the other half by leveraging this symmetry.
The given exercise uses this property to simplify the calculation of the sum for the first half of the series \( \sum_{r=0}^{10} \binom{20}{r} \), by showing that it is equivalent to the sum from \( r=11 \) to \( r=20 \). This leads to a simplified solution, as calculating half the sum directly gives you half of the entire series.

Binomial Expansion

The binomial expansion refers to the expansion of a power of a binomial expression based on the binomial theorem. The binomial theorem states that:\[ (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]Each term in this expansion has a co-efficient \( \binom{n}{k} \), a power of \( a \), and a power of \( b \). These terms outline how one can expand a simple binomial into a sum of terms of different powers of \( a \) and \( b \).
In practical scenarios, such as the given exercise, binomial expansion allows the calculation of sums involving binomial coefficients without expanding each term individually. For example, calculating \( \sum_{r=0}^{20} \binom{20}{r} \) involves using the property of the binomial expansion rather than manually working through each coefficient.
Moreover, the expansion is used to determine specific probabilities and outcomes in various probability-related questions, as each binomial expansion expresses the likelihood of different combinations happening when there are two possible outcomes. Comprehending how the binomial expansion works can simplify complex problems in both algebra and probability.