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A First Course in Probability

Sheldon M. Ross

$Math Studyset Vaia Explanations$ Math

9th Edition · 432 pages

ISBN: 9780321794772

Chapter 1: Q.1.3 (page 17)

In how many ways can $r$ objects be selected from a set of $n$ objects if the order of selection is considered relevant?

Short Answer

Expert verified

Choose the first, then the second, and so on...

$\frac{n!}{(n - r)!}$

Step by step solution

01

Given Information.

$r$ objects are selected from a set of $n$ objects if the order of selection is considered relevant.

02

Explanation.

Choosing $r$ from $n$ , ordered

The first element can be selected in $n$ ways

The second can be any of the remaining $n - 1$

The $r$ element can be any of the $n - r + 1$ elements.

By the generalized principle of the product, there are $n \cdot (n - 1) \cdot \dots (n - r + 1) = \frac{n!}{(n - r)!}$ possible choices.

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Most popular questions from this chapter

Argue that

$(\begin{matrix} n \\ n_{1}, n_{2}, . . . . . . . ., n_{r} \end{matrix}) = (\begin{matrix} n - 1 \\ n_{1} - 1, n_{2}, . . . . . . . ., n_{r} \end{matrix}) + (\begin{matrix} n - 1 \\ n_{1}, n_{2} - 1, . . . . . . . ., n_{r} \end{matrix}) + . . . . . . + (\begin{matrix} n - 1 \\ n_{1}, n_{2}, . . . . . . . ., n_{r} - 1 \end{matrix})$

Hint: Use an argument similar to the one used to establish Equation (4.1).

Present a combinatorial explanation of why

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

A dance class consists of $22$ students, of which $10$ are women and 12 are men. If $5$ men and $5$ women are to be

chosen and then paired off, how many results are possible?

A president, treasurer, and secretary, all different, are to be chosen from a club consisting of $10$ people. How many different choices of officers are possible if

(a) there are no restrictions?

(b) A and B will not serve together?

(c) C and D will serve together or not at all?

(d) E must be an officer?

(e) F will serve only if he is president?

Prove that:

$(\begin{matrix} n + m \\ r \end{matrix}) = (\begin{matrix} n \\ 0 \end{matrix}) (\begin{matrix} m \\ r \end{matrix}) + (\begin{matrix} n \\ 1 \end{matrix}) (\begin{matrix} m \\ r - 1 \end{matrix}) + . . . . . . . . . . + (\begin{matrix} n \\ r \end{matrix}) (\begin{matrix} m \\ 0 \end{matrix})$

Hint: Consider a group of $n$ men and $m$ women. How many groups of size $r$ are possible?

See all solutions

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