Question

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

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

Short answer

The number of vectors$(x_1,.....x_n)$is$144and18$

Step-by-step solution

Step-1 Given Information

Given in the question that,

Code is sequence of $3$digits

The first digit is an integer between $2and9$

Second digit is either $0or1$

Third digit is integer between $1and9$

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 $=0or1(2options)$

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

Total number of possible area codes will be

localid="1650010466242" $$\begin{gathered} =8\times 2\times 9 \\ =144 \end{gathered}$$

Case (b):

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

Second digit$=0or1(2options)$

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

Total number of possible area codes will be

localid="1650010663142" $$\begin{gathered} =1\times 2\times 9 \\ =18 \end{gathered}$$