Chapter 1: Q.1.5 (page 17)

Determine the number of vectors $(x_{1}, . . ., x_{n}),$ such that each $x_{i}$ is either $0$ or $1$ and $\sum_{i = 1}^{n} x_{i} \geq k$ .

Short Answer

Expert verified

There are $\sum_{m = k}^{n} \frac{n!}{m! (n - m)!}$ different ways.

Step by step solution

Given Information.

The number of vectors $(x_{1}, \dots, x_{n})$ is either 0 or 1and $\sum_{i = 1}^{n} x_{i} \geq k$ .

Explanation.

The condition $\sum_{i = 1}^{n} x_{i} \geq k$ holds if and only if the vector $x = (x_{1}, \dots, x_{n})$ has $k$ or more components equal to $1$ .

Let us fix that the vector shall have exactly $m$ components equal to $1$ . Therefore, we can arrange this collection of ones and zeros in $n!$ different ways. But, because we do not want to count twice the permutations of zeros and ones among ourselves, we should divide byrole="math" localid="1648738905041" $m! (m - k)!$

The result follows by considering all $m$ between $k$ and $n$ .

Explanation.

There are $\sum_{m = k}^{n} \frac{n!}{m! (n - m)!}$ different ways.

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Determine the number of vectors $(x_{1}, . . ., x_{n}),$ such that each $x_{i}$ is either $0$ or $1$ and $\sum_{i = 1}^{n} x_{i} \geq k$ .

Short Answer

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Determine the number of vectors (x1,...,xn),such that each xiis either 0or1and∑i=1nxi≥k.

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Determine the number of vectors $(x_{1}, . . ., x_{n}),$ such that each $x_{i}$ is either $0$ or $1$ and $\sum_{i = 1}^{n} x_{i} \geq k$ .