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" $\sum_{i = 1}^{n} x_{i} \leq k$

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 $\sum_{p = 0}^{k} (\begin{matrix} p + n - 1 \\ n - 1 \end{matrix})$

Therefore,

$\sum_{p = 0}^{k} (\begin{matrix} p + n - 1 \\ n - 1 \end{matrix}) = (\begin{matrix} n - 1 \\ n - 1 \end{matrix}) + (\begin{matrix} n \\ n - 1 \end{matrix}) + (\begin{matrix} n + 1 \\ n + 1 \end{matrix}) + . . . . . . . + (\begin{matrix} k + n - 1 \\ n - 1 \end{matrix}) \Rightarrow \sum_{p = 0}^{k} (\begin{matrix} p + n - 1 \\ n - 1 \end{matrix}) = 1 + (\begin{matrix} n \\ n - 1 \end{matrix}) + (\begin{matrix} n \\ n \end{matrix}) + (\begin{matrix} n + 1 \\ n + 1 \end{matrix}) + . . . . . . . + (\begin{matrix} k + n - 1 \\ n - 1 \end{matrix}) - (\begin{matrix} n \\ n \end{matrix}) \Rightarrow \sum_{p = 0}^{k} (\begin{matrix} p + n - 1 \\ n - 1 \end{matrix}) = 1 + (\begin{matrix} k + n \\ n \end{matrix}) - 1 \Rightarrow \sum_{p = 0}^{k} (\begin{matrix} p + n - 1 \\ n - 1 \end{matrix}) = (\begin{matrix} k + n \\ n \end{matrix})$

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

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

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find the number of vectors.

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

Geometry

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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" $\sum_{i = 1}^{n} x_{i} \leq k$