Chapter 13.1, Problem 18E

a.

To determine

To calculate: The number of different 7-digit telephone number if first digit cannot be 0.

Answer to Problem 18E

The number of different 7-digit telephone number if first digit cannot be 0are $9000000$ .

Explanation of Solution

Given information:

7-digit telephone numbers are to be constructed and first digit cannot be zero.

Calculation:

Consider the provided information that 7-digit telephone numbers are to be constructed and first digit cannot be zero.

Let 7 digit telephone number be $\boxed{}\boxed{}\boxed{}\boxed{}\boxed{}\boxed{}\boxed{}$ .

Digits available are from 0 to 9.

Since, first digit cannot be 0 so there are 9 ways to fill the first box.

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

And for remaining boxes there are 10 choices for each.

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

Therefore, the possible ways are $9\cdot {10}^6=9000000$ .

Thus, the number of different 7-digit telephone number if first digit cannot be 0 are $9000000$ .

b.

To determine

To calculate: The number of different 7-digit telephone number if only even digits are to be used.

Answer to Problem 18E

The number of different 7-digit telephone number if only even digits are to be usedare $78125$ .

Explanation of Solution

Given information:

7-digit telephone numbers are to be constructed and only even digits are to be used.

Calculation:

Consider the provided information that 7-digit telephone numbers are to be constructed and only even digits are to be used.

Let 7 digit telephone number be $\boxed{}\boxed{}\boxed{}\boxed{}\boxed{}\boxed{}\boxed{}$ .

Digits available are 0, 2, 4, 6 and 8.

Since, only even digits are to be used so there are 5 ways to fill the first box.

  $\boxed{5}\boxed{}\boxed{}\boxed{}\boxed{}\boxed{}\boxed{}$

And for remaining boxes there are 5 choices for each.

  $\boxed{5}\boxed{5}\boxed{5}\boxed{5}\boxed{5}\boxed{5}\boxed{5}$

Therefore, the possible ways are $5^7=78125$ .

Thus, the number of different 7-digit telephone number if only even digits are to be usedare $78125$ .

c.

To determine

To calculate: The number of different 7-digit telephone number if the complete number is a multiple of 10.

Answer to Problem 18E

The number of different 7-digit telephone number if the complete number is a multiple of 10are $1000000$ .

Explanation of Solution

Given information:

7-digit telephone numbers are to be constructed and the complete number is a multiple of 10.

Calculation:

Consider the provided information that 7-digit telephone numbers are to be constructed and the complete number is a multiple of 10.

Let 7 digit telephone number be $\boxed{}\boxed{}\boxed{}\boxed{}\boxed{}\boxed{}\boxed{}$ .

Digits available are from 0 to 9.

Since, the complete number is a multiple of 10so there are 1 ways to fill the last box that is digit can be 0.

  $\boxed{}\boxed{}\boxed{}\boxed{}\boxed{}\boxed{}\boxed{0}$

And for remaining boxes there are 10 choices for each.

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

Therefore, the possible ways are ${10}^6\cdot 1=1000000$ .

Thus, the number of different 7-digit telephone number if the complete number is a multiple of 10 are $1000000$ .

d.

To determine

To calculate: The number of different 7-digit telephone number if the first three digits are 593 in an order.

Answer to Problem 18E

The number of different 7-digit telephone number if the first three digits are 593 in an orderare $10000$ .

Explanation of Solution

Given information:

7-digit telephone numbers are to be constructed and the first three digits are 593 in an order.

Calculation:

Consider the provided information that 7-digit telephone numbers are to be constructed and the first three digits are 593 in an order.

Let 7 digit telephone number be $\boxed{}\boxed{}\boxed{}\boxed{}\boxed{}\boxed{}\boxed{}$ .

Digits available are from 0 to 9.

Since, the first three digits are 593 in an order so there is 1 way to fill the digit that is with 5, 1 way to fill the second digit that is with 9 and 1 way to fill the third digit that is with 3.

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

And for remaining boxes there are 10 choices for each.

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

Therefore, the possible ways are $1\cdot {10}^4=10000$ .

Thus, the number of different 7-digit telephone number if the first three digits are 593 in an order are $10000$ .