Q. 1.7 Give a combinatorial explanation... [FREE SOLUTION]

91影视

A First Course in Probability

Sheldon M. Ross

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 432 pages

ISBN: 9780321794772

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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91影视

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.

One App. One Place for Learning.

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91影视

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

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Study anywhere. Anytime. Across all devices.

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