/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 20 Find the value of each combinati... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 20

Find the value of each combination. $$ { }_{9} C_{2} $$

Short Answer

Expert verified
The value of \[ { }_{9} C_{2} \]is 36.

Step by step solution

01

Understand the Combination Formula

The combination formula to find the number of ways to choose 'r' elements from a set of 'n' elements is given by \[ { }_{n} C_{r} = \frac{n!}{r!(n-r)!} \]
02

Substitute the Given Values

In the given problem, we need to find \[ { }_{9} C_{2} \]Here, \( n = 9 \) and \( r = 2 \). Substitute these values into the formula: \[ { }_{9} C_{2} = \frac{9!}{2!(9-2)!} \]
03

Simplify the Factorials

Simplify the factorial terms in the formula. Start by writing the factorials: \[ 9! = 9 \times 8 \times 7! \]Therefore: \[ { }_{9} C_{2} = \frac{9 \times 8 \times 7!}{2! \times 7!} \]
04

Cancel Common Factors

As you can see, \( 7! \) in the numerator and denominator cancel out: \[ { }_{9} C_{2} = \frac{9 \times 8}{2!} \]
05

Calculate the Remaining Terms

Now, calculate the value of \( 2! \): \[ 2! = 2 \times 1 = 2 \]Hence: \[ { }_{9} C_{2} = \frac{9 \times 8}{2} = \frac{72}{2} = 36 \]

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Key Concepts

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

Factorials
Factorials are a foundational concept in combinatorics and various mathematical calculations. The factorial of a non-negative integer, denoted as \(!n\), is the product of all positive integers less than or equal to \(!n\). For example, \[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]. Factorials are used in permutations, combinations, and other areas of mathematics. Understanding how to simplify and manipulate factorials can greatly assist in solving problems involving large numbers.
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Most popular questions from this chapter

What is the probability of an event that is impossible? Suppose that a probability is approximated to be zero based on empirical results. Does this mean the event is impossible?

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