Question

All 5-digit zip codes in the United States are assigned by region. In one particularstate, all zip codes begin with the number 3 while the second digit is restricted tonumbers from 3 to 7. There is no restriction on the remaining digits. How manyzip code combinations are possible in this particular state? (8 points)

Answer 1

We want to determine the number of zip code combinations possible for the state.

$\begin{gathered} N=N_1\times N_2\times N_3\times N_4\times N_5 \\ \\ \text{Where;} \\ N\text{ is the total number of zip code combination possible for the state.} \\ N_1\ldots N_{5\text{ }}are\text{ the possible numbers that can fill into each digit of the code from the first to the last.} \end{gathered}$

Given that;

- all zip codes begin with the number 3

- the second digit is restricted to numbers from 3 to 7

- the remaining digits have no restriction.

With the above conditions;

the first digit can only be one number which is 3, so;

$N_1=1$

the second digit can be just 2 numbers either 3 or 7, so;

$N_2=2$

The remaining digits have 10 possible numbers each from 0 to 9, so;

$N_3=N_4=N_5=10$

substituting this values into the formula given, we have;

$\begin{gathered} N=N_1\times N_2\times N_3\times N_4\times N_5 \\ N=1\times2\times10\times10\times10 \\ N=2000 \end{gathered}$

Therefore, the state have 2000 possible zip code combinations.