Question

\(\sum_{r=2}^{16}{ }^{16} \mathrm{C}_{r}=\) (1) \(2^{15}-15\) (2) \(2^{16}-16\) (3) \(2^{16}-17\) (4) \(2^{17}-17\)

Short answer

Answer: (3) \(2^{16}-17\).

Step-by-step solution

Write the Expression for Sum of Combinations

We want to compute the sum of the combinations, given as: \(\sum_{r=2}^{16}{ }^{16} \mathrm{C}_{r}\).

Simplify the Expression using Binomial Theorem

We know from binomial theorem, \((1 + x)^{n} = \sum_{r=0}^{n}{ }^{n}\mathrm{C}_{r}x^r\) In our case, we can rewrite the expression : \(\sum_{r=2}^{16}{ }^{16} \mathrm{C}_{r} = (\sum_{r=0}^{16}{ }^{16}\mathrm{C}_{r}) - \,^{16}\mathrm{C}_{0} - \,^{16}\mathrm{C}_{1}\) Put n=16 and x=1 in the binomial theorem, we get: \((1+1)^{16} = \sum_{r=0}^{16}\,^{16}\mathrm{C}_{r}1^r = 2^{16}\)

Subtract the Extra Terms and Evaluate the Expression

Substitute the result of the binomial theorem back into the expression derived in step 2: \(\sum_{r=2}^{16}{ }^{16} \mathrm{C}_{r} = 2^{16} - \,^{16}\mathrm{C}_{0} - \,^{16}\mathrm{C}_{1}\) We know that \(\,^{n}\mathrm{C}_{0} = 1\) and \(\,^{n}\mathrm{C}_{1} = n\) So, \(\sum_{r=2}^{16}{ }^{16} \mathrm{C}_{r} = 2^{16} - 1 - 16\)

Final Answer

Calculate the expression: \(\sum_{r=2}^{16}{ }^{16} \mathrm{C}_{r} = 2^{16} - 1 - 16 = 2^{16} - 17\) Thus, the correct answer is (3) \(2^{16}-17\).

Key concepts

Binomial Theorem

The Binomial Theorem is a fundamental principle in combinatorics, which provides a powerful way to expand expressions of the form \((a + b)^n\). It states that:

• \((a + b)^n = \sum_{r=0}^{n} {^n C_r} a^{n-r} b^r\), where \({^n C_r}\) is a binomial coefficient.
• This coefficient, \({^n C_r}\), represents the number of ways to choose \(r\) items from \(n\) without regard to order.

The theorem simplifies calculations by allowing the expression to be expanded into a series that is computationally manageable. It plays a significant role in algebra and probability, among other fields. In our problem, by choosing appropriate values for \(a\) and \(b\), we effectively used the theorem to simplify a sum of combinations.

Mathematical Proofs

Mathematical proofs are logical arguments that demonstrate the truth of a statement. They serve as the foundation for mathematical logic and ensure consistency in mathematical theory.

• Proofs often begin with known truths or axioms and proceed through logical steps to arrive at a conclusion.
• In our problem, we used the Binomial Theorem as an axiom to deduce the value of the given sum of combinations.

Knowing how to construct a proof is an essential skill for solving mathematical problems because it ensures that solutions are not only found but are also accurate and reliable. Breaking complicated problems into smaller, more manageable parts is key to forming an effective proof.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operators (like +, -, *, /). They represent values and can describe complex mathematical ideas in a compact form. Here are some key points:

• Expressions can be simplified using known formulas and rules, such as the Binomial Theorem, which was applied in our problem to manage a sum of combinations.
• Understanding the components of expressions and how they interact is crucial for manipulating and simplifying them properly.

In the original exercise, we started with the expression \(\sum_{r=2}^{16}{ }^{16} \mathrm{C}_{r}\) and applied mathematical principles to simplify it. Correct execution of steps like these ensures accurate problem-solving in mathematics.