Chapter 1: Q. 1.7 (page 19)

Give a combinatorial explanation of the identity
$(\begin{matrix} n \\ r \end{matrix}) = (\begin{matrix} n \\ n - r \end{matrix})$

Short Answer

Expert verified

The given identity $(\begin{matrix} n \\ r \end{matrix}) = (\begin{matrix} n \\ n - r \end{matrix})$ can be explained as the no. of possible combinations of selecting r objects from n objects is equivalent to selecting $n - r$ objects from $n$ objects.

Step by step solution

Step 1. Given information.

The given identity is

$(\begin{matrix} n \\ r \end{matrix}) = (\begin{matrix} n \\ n - r \end{matrix})$

and we have to give its combinatorial explanation.

We know that $(\begin{matrix} x \\ r \end{matrix}) = \frac{x!}{r! (x - r)!}$ .

Step 2. Give combinatorial explanation of the given identity.

Taking L.H.S of the given identity, we have

$(\begin{matrix} n \\ r \end{matrix})$ , which can be written as $\frac{n!}{r! (n - r)!}$ .

Therefore, $(\begin{matrix} n \\ r \end{matrix}) = \frac{n!}{r! (n - r)!}$

Taking R.H.S of the given identity, we have $(\begin{matrix} n \\ n - r \end{matrix})$ , which can be written as $\frac{n!}{(n - r)! (n - n + r)!} = \frac{n!}{r! (n - r)!}$

Therefore, $(\begin{matrix} n \\ n - r \end{matrix}) = \frac{n!}{r! (n - r)!}$

Hence, it is proved that $(\begin{matrix} n \\ r \end{matrix}) = (\begin{matrix} n \\ n - r \end{matrix})$

The given identity role="math" localid="1648811341364" $(\begin{matrix} n \\ r \end{matrix}) = (\begin{matrix} n \\ n - r \end{matrix})$ can be explained as the no. of possible combinations of selecting $r$ objects from $n$ objects is equivalent to selecting $n - r$ objects from $n$ objects.

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Give a combinatorial explanation of the identity
$(\begin{matrix} n \\ r \end{matrix}) = (\begin{matrix} n \\ n - r \end{matrix})$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Give combinatorial explanation of the given identity.

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Give a combinatorial explanation of the identitynr=nn-r

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Give combinatorial explanation of the given identity.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Theoretical and Mathematical Physics

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Study anywhere. Anytime. Across all devices.

Give a combinatorial explanation of the identity
$(\begin{matrix} n \\ r \end{matrix}) = (\begin{matrix} n \\ n - r \end{matrix})$