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Chapter 9: Problem 6

In Exercises 5 - 14, calculate the binomial coefficient. \( _8C_6 \)

Short Answer

Expert verified
The binomial coefficient \(_8C_6\) is 28.

Step by step solution

01

Calculate Factorials

First, calculate the factorials involved in the formula. That is, calculate \(8!\), \(6!\), and \(2!\).\n\n 8! = 8*7*6*5*4*3*2*1 = 40320\n 6! = 6*5*4*3*2*1 = 720\n 2! = 2*1 = 2
02

Apply Factorials to Formula

Next, apply the calculated factorial values to the binomial coefficient formula. \(_nC_k = \frac{n!}{k!(n-k)!}\) becomes \(_8C_6 = \frac{8!}{6!(8-6)!}\ = \frac{40320}{720 * 2}\.
03

Perform Division

Finally, perform the division operation as indicated in the formula. \(_8C_6 = \frac{40320}{1440} = 28\).

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Key Concepts

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

Understanding Factorials
Factorials are fundamental in many areas of mathematics, especially in permutations, combinations, and binomial coefficients. A factorial, denoted by an exclamation mark (!), of a non-negative integer is the product of all positive integers less than or equal to that number. For example, the factorial of 4, written as \(4!\), is \(4 \times 3 \times 2 \times 1 = 24\).

Here are some useful properties of factorials:
  • \(0! = 1\): By definition, the factorial of zero is one.
  • For any whole number \(n\), \(n!\) is \(n \times (n-1)!\).
  • Factorials grow very quickly, with the value increasing tremendously as \(n\) increases.
Understanding how to compute factorials is key to solving problems involving binomial coefficients.
Exploring Combinatorics
Combinatorics is a field of mathematics concerned with counting, arranging, and finding patterns. It involves studying various ways in which discrete structures can be combined or structured.

One of the most common scenarios in combinatorics involves calculating the number of ways to choose \(k\) items from a larger set of \(n\) distinct items. This calculation is known as a combination. The formula for combinations, also known as the binomial coefficient, is \(_nC_k = \frac{n!}{k!(n-k)!}\), where \(n!\) is the factorial of \(n\), and so on.

Key points in combinatorics:
  • Order does not matter in combinations, unlike permutations where order is significant.
  • Combinatorics is essential in probability, statistics, and computer science for analyzing different scenarios.
Applying the Binomial Theorem
The Binomial Theorem provides a quick way to expand powers of binomials. A binomial is simply a two-term expression like \((a + b)\). The theorem states that for any positive integer \(n\), the expansion of \((a + b)^n\) is given by:
  • \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k}b^{k}\)
Here, \(\binom{n}{k}\) denotes the binomial coefficient, which tells us the number of ways to choose \(k\) terms from \(n\), and is equivalent to \(_nC_k\).

Application of the Binomial Theorem:
  • This theorem helps simplify the process of raising binomials to a power without directly multiplying them.
  • Binomial coefficients, or combinations, are integral within the expansion to determine coefficients of each term.
By understanding the binomial coefficient and the theorem itself, solving polynomial expansion problems becomes much simpler.

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Most popular questions from this chapter

In Exercises 35 - 38, you are given the probability that an event will happen. Find the probability that the event will not happen. \( P(E) = \dfrac{2}{3} \)

In Exercises 21 - 24, find the probability for the experiment of selecting one card from a standard deck of \( 52 \) playing cards. The card is a \( 9 \) or lower. (Aces are low.)

A police department uses computer imaging to create digital photographs of alleged perpetrators from eyewitness accounts. One software package contains \( 195 \) hairlines, \( 99 \) sets of eyes and eyebrows, \( 89 \) noses, \( 105 \) mouths, and \( 74 \) chins and cheek structures. (a) Find the possible number of different faces that the software could create. (b) An eyewitness can clearly recall the hairline and eyes and eyebrows of a suspect. How many different faces can be produced with this information?

In Exercises 21 - 24, find the probability for the experiment of selecting one card from a standard deck of \( 52 \) playing cards. The card is a red face card.

A fire company keeps two rescue vehicles. Because of the demand on the vehicles and the chance of mechanical failure, the probability that a specific vehicle is available when needed is \( 90\% \). The availability of one vehicle is independent of the availability of the other. Find the probabilities that (a) both vehicles are available at a given time, (b) neither vehicle is available at a given time, and (c) at least one vehicle is available at a given time.
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