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Chapter 6: Problem 4

Evaluate the binomial coefficient. $$\left(\begin{array}{c} 10 \\ 5 \end{array}\right)$$

Short Answer

Expert verified
The value of \(\binom{10}{5}\) is 252.

Step by step solution

01

Substitute into formula

The first step requires to put the numbers 10 and 5 into the binomial coefficient formula. We get \[ \binom{10}{5} = \frac{10!}{5!(10-5)!} \]
02

Simplify the factorial expressions

Factorials are generally complex to compute; the factorial n! is defined as the product of all positive integers less than or equal to n. So, here it is necessary to simplify the expression to get the desired result. We get \[ \binom{10}{5} = \frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2 \times 1} \]
03

Cancel terms

Cancelling terms simplifies the process of calculation. We observe that 10 = 5 * 2, 8 = 4 * 2, 9/3 = 3, and 6/2 = 3, so we can actually cancel terms from the numerator and denominator. This leaves us with \[ \binom{10}{5} = 2 \times 3 \times 7 = 42 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factorials
Factorials are a fundamental concept in mathematics. They are denoted by the exclamation mark symbol '!'. A factorial of a number, say 'n', denoted as 'n!', is defined as the product of all positive integers up to 'n'.
  • For example, the factorial of 5, expressed as 5!, is 5 脳 4 脳 3 脳 2 脳 1 = 120.
  • Factorial expressions grow rapidly larger as the value of n increases.
  • For instance, 10! equates to 3,628,800, showing how quickly values intensify.
Applications: Factorials are not just theoretical. They are crucial in various areas such as probability, algebra, and number theory.
In the Binomial Coefficient: They appear in computations of combinations, which is a key concept in binomial coefficients.
Combinatorics
Combinatorics is the field of mathematics concerning the counting, arrangement, and combination of objects.It is immensely useful in solving problems related to probability, algorithms, and many real-world scenarios.
  • The binomial coefficient, denoted as \( \binom{n}{k} \), is a prominent concept in combinatorics.
  • It represents the number of ways to choose 'k' elements from a set of 'n' elements without regard to order.
Example: Consider \( \binom{10}{5} \), which means choosing 5 objects from a total of 10. This calculation helps in situations like forming teams, delegating tasks, or distributing prizes. Notice how central factorials are used here, emphasizing the interconnectedness of these concepts.
Mathematical Simplification
Simplifying mathematical expressions is a vital skill. It makes calculations manageable and results more understandable. When computing high numbers or lengthy equations, identifying common terms is essential.
In the context of our problem, simplifying the Binomial Coefficient involves breaking down:
  • Canceling common factors in the numerator and denominator;
  • Recognizing equivalent terms that cancel out some factorial expressions.
Key Meanings: In our example \( \binom{10}{5} \), it involves realizing that several factors can cancel.For instance, some numbers in 10! and 5! mirror each other, creating cancellations like \( 10 = 5 \times 2 \).These operations speed up solving problems without having to fully compute large numbers.

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Most popular questions from this chapter

If the sequence \(a_{n}\) is an arithmetic sequence, make a conjecture about the sequence \(2^{a_{n}}\) and give a proof.

Evaluate \(5 !\)

Solve \(y=\ln t\) for \(t .[3.5]\)

Write each rational number as the quotient of two integers in simplest form. $$0 . \overline{123}$$

Use the Binomial Theorem to simplify the powers of the complex numbers. $$(3+2 i)^{3}$$
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