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Chapter 14: Problem 8

Fill in each blank with the correct response. For any nonnegative integer \(n,\) the binomial coefficient \({ }_{n} C_{n}\) is equal to ______.

Short Answer

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Step by step solution

01

Understanding Binomial Coefficient

The binomial coefficient \({ }_{n} C_{n}\) represents the number of ways to choose \( n \) elements from \( n \) elements. It is also read as 'n choose n'.
02

Using the Binomial Coefficient Formula

The general formula for the binomial coefficient is given by: \(\binom{n}{k} = \frac{n!}{k!(n-k)!} \). For \({ }_{n} C_{n}\), substitute \( k = n \).
03

Simplifying the Expression

Substitute \( k = n \) in the formula: \({ }_{n} C_{n} = \frac{n!}{n!(n-n)!} = \frac{n!}{n! \times 0!}\).
04

Evaluating Factorials

Recall that \( 0! = 1 \). Therefore: \(\frac{n!}{n! \times 1} = \frac{n!}{n!} = 1 \).

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Key Concepts

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

factorials
A factorial is a function represented by an exclamation mark (!) after a number. It multiplies that number by all the smaller natural numbers down to 1. This means that for a positive integer , the factorial of is calculated as follows:

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Most popular questions from this chapter

Fill in each blank with the correct response. For any nonnegative integer \(n,\) the binomial coefficient \({ }_{n} C_{0}\) is equal to ______.

In a certain community, the consumption of electricity has increased about \(6 \%\) per yr. (a) If the community uses 1.1 billion units of electricity now, how much will it use 5 yr from now? Round to the nearest tenth. (b) Find the number of years (to the nearest year) it will take for the consumption to double.

Write the first five terms of each sequence. $$ a_{n}=5(-1)^{n-1} $$

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Fill in each blank with the correct response. The arithmetic mean of \(-4,-2,0,2,\) and 4 is ____.
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