Q. 1.23 Determine the number of vectors ... [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. 1.23 (page 19)

Determine the number of vectors $(x 1, . . ., x n)$ such that each $x_{i}$ is a nonnegative integer and localid="1648403251141" ${鈭�}_{i = 1}^{n} x_{i} 鈮� k$

Short Answer

Expert verified

The number of vectors are $(\begin{matrix} k + n \\ n \end{matrix})$ .

Step by step solution

Step 1. Given information.

From the given information, we can observe that the number of vectors $(x_{1}, x_{2} . . ., x_{n})$ such that each $x_{i}$ is non negative integer.

Step 2. Find the number of vectors.

The number of vectors will be represented by ${鈭�}_{p = 0}^{k} (\begin{matrix} p + n - 1 \\ n - 1 \end{matrix})$

Therefore,

${鈭�}_{p = 0}^{k} (\begin{matrix} p + n - 1 \\ n - 1 \end{matrix}) = (\begin{matrix} n - 1 \\ n - 1 \end{matrix}) + (\begin{matrix} n \\ n - 1 \end{matrix}) + (\begin{matrix} n + 1 \\ n + 1 \end{matrix}) + . . . . . . . + (\begin{matrix} k + n - 1 \\ n - 1 \end{matrix}) 鈬� {鈭�}_{p = 0}^{k} (\begin{matrix} p + n - 1 \\ n - 1 \end{matrix}) = 1 + (\begin{matrix} n \\ n - 1 \end{matrix}) + (\begin{matrix} n \\ n \end{matrix}) + (\begin{matrix} n + 1 \\ n + 1 \end{matrix}) + . . . . . . . + (\begin{matrix} k + n - 1 \\ n - 1 \end{matrix}) - (\begin{matrix} n \\ n \end{matrix}) 鈬� {鈭�}_{p = 0}^{k} (\begin{matrix} p + n - 1 \\ n - 1 \end{matrix}) = 1 + (\begin{matrix} k + n \\ n \end{matrix}) - 1 鈬� {鈭�}_{p = 0}^{k} (\begin{matrix} p + n - 1 \\ n - 1 \end{matrix}) = (\begin{matrix} k + n \\ n \end{matrix})$

So, the number of vectors are $(\begin{matrix} k + n \\ n \end{matrix})$ .

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91影视

Determine the number of vectors $(x 1, . . ., x n)$ such that each $x_{i}$ is a nonnegative integer and localid="1648403251141" ${鈭�}_{i = 1}^{n} x_{i} 鈮� k$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find the number of vectors.

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91影视

Determine the number of vectors (x1,...,xn)such that each xiis a nonnegative integer and localid="1648403251141" 鈭�i=1nxi鈮�k

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find the number of vectors.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

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Mechanics Maths

Probability and Statistics

Discrete Mathematics

Calculus

Study anywhere. Anytime. Across all devices.

Determine the number of vectors $(x 1, . . ., x n)$ such that each $x_{i}$ is a nonnegative integer and localid="1648403251141" ${鈭�}_{i = 1}^{n} x_{i} 鈮� k$