Hint: The formula used to find the number of different combinations of n distinct elements taken r at a time is $^n{C_r} = \dfrac{{n!}}{{r!(n - r)!}}$ . By substituting the values from the question in the expression, we can find the value of ${}^7{C_2}$.
Complete Step by Step Solution:
We need to evaluate the given expression to find the possible combinations for choosing 2 elements from 7 distinct elements. We need to keep in mind that we need not consider the order in which these elements appear when chosen.
The formula we use to evaluate an expression of the form $^n{C_r}$ is
${ \Rightarrow ^n}{C_r} = \dfrac{{n!}}{{r!(n - r)!}}$
From the question, we have n=7 and r=2.
Substituting the values of n and r in the formula
$ \Rightarrow {}^7{C_2} = \dfrac{{7!}}{{2!(7 - 2)!}} = \dfrac{{7!}}{{2!(5)!}}$
We further simply the factorial notations,
7! Can be expressed as $7! = 7 \times 6 \times 5!$ . Therefore,
$ \Rightarrow {}^7{C_2} = \dfrac{{7 \times 6 \times 5!}}{{(2 \times 1)(5)!}}$
Cancelling common terms in the above expression, we get
$ \Rightarrow {}^7{C_2} = \dfrac{{7 \times 6}}{{2 \times 1}}$
Simplifying the expression further,
$ \Rightarrow {}^7{C_2} = \dfrac{{42}}{2} = 21$
Hence, the value of the expression ${}^7{C_2}$ is 21. This means that there are 21 combinations for choosing 2 elements from 7 distinct elements.
Note:
The factorial function which is represented by the symbol “!” simply means that we need to multiply a series of descending natural numbers. For example, $4! = 4 \times 3 \times 2 \times 1$. The value of 0! Is considered to be 1.
As a seasoned mathematician and enthusiast in combinatorics, I can confidently guide you through the step-by-step solution provided in the article, showcasing my depth of knowledge in this field.
Firstly, the article introduces the formula for finding the number of different combinations of (n) distinct elements taken (r) at a time as (^nC_r = \dfrac{n!}{r!(n - r)!}). This formula is a fundamental concept in combinatorics, specifically addressing the number of ways to choose (r) elements from a set of (n) distinct elements without considering their order.
Let's break down the steps outlined in the article to find the value of (^7C_2):
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Identification of Parameters: The article specifies that we have (n = 7) and (r = 2). This information is crucial for substituting values into the combination formula.
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Application of Combination Formula: By substituting (n = 7) and (r = 2) into the combination formula, we get (^7C_2 = \dfrac{7!}{2!(7 - 2)!}).
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Factorial Notation: The article proceeds to simplify the factorial notation. (7!) is expressed as (7 \times 6 \times 5!). This step is a common technique to break down factorials into smaller terms.
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Cancellation of Common Terms: The expression is further simplified by canceling common terms in the numerator and denominator, resulting in (^7C_2 = \dfrac{7 \times 6}{2 \times 1}).
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Final Simplification: The last step involves simplifying the expression to obtain (^7C_2 = \dfrac{42}{2} = 21).
In conclusion, the value of (^7C_2) is determined to be 21, indicating that there are 21 combinations for choosing 2 elements from a set of 7 distinct elements.
The article also provides a useful note about the factorial function, denoted by "!" and explains that (n!) involves multiplying a series of descending natural numbers. For instance, (4! = 4 \times 3 \times 2 \times 1), and (0!) is considered to be 1. This note adds clarity to the understanding of factorials, a concept integral to combinatorics and probability theory.
