Q.1.5 - Theoretical Exercises - A First Course in Probability

Question

Determine the number of vectors$(x_1,...x_{n)}$  such that each $xi$ is either $0$ or $1$ and

$\sum _{i=1}^n{x}_i\ge k$

Step-by-Step Solution

Answer

The number of vectors $(x_1,.....x_n)$ is$144 and 18$

Step-1 Given Information

Given in the question that,

Code is sequence of $3$ digits

The first digit is an integer between $2 and 9$

Second digit is either $0 or 1$

Third digit is integer between $1 and 9$

we have to find the number of vectors.

Step-2 Explanation

Case (a):$3$digit code

First digit $=8$ options $\left(2-9\right)$

Second digit $=0 or 1 (2 options)$

Third digit $=9$options$(1-9)$

Total number of possible area codes will be

$$\begin{gathered} =8\times 2\times 9 \\ =144 \end{gathered}$$       

Case (b):

First digit  $=1$options $\left(4\right)$

Second digit $=0 or 1 (2 options)$

Third digit $=9$options$(1-9)$

Total number of possible area codes will be 

$$\begin{gathered} =1\times 2\times 9 \\ =18 \end{gathered}$$