Chapter 13.1, Problem 15CFU

a.

To determine

To calculate: The number of different 5-digit ZIP codes.

Answer to Problem 15CFU

The number of different 5-digit ZIP codes are $100000$ .

Explanation of Solution

Given information:

5-digit ZIP codes is to be constructed with digits from 0 to 9.

Calculation:

Consider the provided information that 5-digit security code is to be constructed with digits from 0 to 9.

Let 5 digit telephone number be $\boxed{}\boxed{}\boxed{}\boxed{}\boxed{}$ .

Digits available are from 0 to 9.

Since, there are no restriction on arrangement of digits so there are 10 ways to fill the first digit.

  $\boxed{10}\boxed{}\boxed{}\boxed{}\boxed{}$

And for remaining boxes also there are 10 choices for each.

  $\boxed{10}\boxed{10}\boxed{10}\boxed{10}\boxed{10}$

Therefore, the possible ways are ${10}^5=100000$ .

Thus, the number of different 5-digit security code are $100000$ .

b.

To determine

To calculate: The number of different 5-digit ZIP codes if first digit is 0 and second, third and fourth digits cannot be 0.

Answer to Problem 15CFU

The number of different 5-digit ZIP codes if first digit is 0 and second, third and fourth digits cannot be 0 are $7290$ .

Explanation of Solution

Given information:

5-digit ZIP codes is to be constructed with digits from 0 to 9 and if first digit is 0 and second, third and fourth digits cannot be 0.

Calculation:

Consider the provided information that 5-digit security code is to be constructed with digits from 0 to 9 and if first digit is 0 and second, third and fourth digits cannot be 0.

Let 5 digit telephone number be $\boxed{}\boxed{}\boxed{}\boxed{}\boxed{}$ .

Digits available are from 0 to 9.

Since, first digit is 0 so there is only 1 way to fill the first digit.

  $\boxed{1}\boxed{}\boxed{}\boxed{}\boxed{}$

Since second, third and fourth digits cannot be 0 so for remaining boxes also there are 9 choices for each.

  $\boxed{1}\boxed{9}\boxed{9}\boxed{9}\boxed{}$

Since fifth digit can be any number from 0 to 9 so there are 10 choices for the box.

  $\boxed{1}\boxed{9}\boxed{9}\boxed{9}\boxed{10}$

Therefore, the possible ways are $1\cdot 9\cdot 9\cdot 9\cdot 10=7290$ .

Thus, the number of different 5-digit ZIP codes if first digit is 0 and second, third and fourth digits cannot be 0 are $7290$ .

c.

To determine

To calculate: The number of additional ZIP codes when 4 more digits were added.

Answer to Problem 15CFU

The number of additional ZIP codes when 4 more digits were addedare $999,900,000$ .

Explanation of Solution

Given information:

5-digit ZIP codes is to be constructed with digits from 0 to 9. Then 4 more digits were added to the existing codes.

Calculation:

Consider the provided information that 5-digit security code is to be constructed with digits from 0 to 9.Then 4 more digits were added to the existing codes.

Let 5 digit telephone number be $\boxed{}\boxed{}\boxed{}\boxed{}\boxed{}$ .

Digits available are from 0 to 9.

Since, there are no restriction on arrangement of digits so there are 10 ways to fill the first digit.

  $\boxed{10}\boxed{}\boxed{}\boxed{}\boxed{}$

And for remaining boxes also there are 10 choices for each.

  $\boxed{10}\boxed{10}\boxed{10}\boxed{10}\boxed{10}$

Therefore, the possible ways are ${10}^5=100000$ .

Now, when 4 more digits were added, a 9 digit ZIP code was there,

Let 9 digit telephone number be $\boxed{}\boxed{}\boxed{}\boxed{}\boxed{}\boxed{}\boxed{}\boxed{}\boxed{}$ .

Digits available are from 0 to 9.

Since, there are no restriction on arrangement of digits so there are 10 ways to fill the first digit.

  $\boxed{10}\boxed{}\boxed{}\boxed{}\boxed{}\boxed{}\boxed{}\boxed{}\boxed{}$

And for remaining boxes also there are 10 choices for each.

  $\boxed{10}\boxed{10}\boxed{10}\boxed{10}\boxed{10}\boxed{10}\boxed{10}\boxed{10}\boxed{10}$

Therefore, the possible ways are ${10}^9=1000000000$ .

So, number of additional codes are,

  $1000000000-100000=999,900,000$

Thus, the number of additional ZIP codes when 4 more digits were added are $999,900,000$ .