Chapter 13.1, Problem 46E

a.

To determine

To calculate: The number of different 5-digit security code.

Answer to Problem 46E

The number of different 5-digit security codeare $100000$ .

Explanation of Solution

Given information:

5-digit security code 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 security codes if no digit is repeated.

Answer to Problem 46E

The number of different 5-digit security codes if no digit is repeatedare $30240$ .

Explanation of Solution

Given information:

5-digit security code 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, no digit is to be repeated there are 10 ways to fill the first digit.

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

For the second digit there are 9 options to choose from because one is already used.

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

For the third digit there are 8 options to choose from because two are already used.

  $\boxed{10}\boxed{9}\boxed{8}\boxed{}\boxed{}$

And similarly,

  $\boxed{10}\boxed{9}\boxed{8}\boxed{7}\boxed{6}$

Therefore, the possible ways are $10\cdot 9\cdot 8\cdot 7\cdot 6=30240$ .

Thus, the number of different 5-digit security codes if no digit is repeated are $30240$ .

c.

To determine

To calculate: The number of different 5-digit security codes if 0 is not allowed and only two digits can be odd and repetition of digits are allowed. Also the number of codes when repetition is not allowed.

Answer to Problem 46E

The number of different 5-digit security codes when repetition of digits is allowed are $1600$ and when repetition of digits is not allowed are 480.

Explanation of Solution

Given information:

5-digit security code 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 9.0 is not allowed and only two digits can be odd and repetition of digits are allowed.

As only 2 digits need to be odd so 3 digits will be even.

There are 5 odd and 4 even numbers between 1 and 9.

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

There are 5 ways to fill the first box and second box.

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

For the third, fourth and fifth box 4 options are there

  $\boxed{5}\boxed{5}\boxed{4}\boxed{4}\boxed{4}$

Therefore, the possible ways are $5\cdot 5\cdot 4\cdot 4\cdot 4=1600$ .

Now, if repetition of digits is not allowed.

There are 5 ways to fill the first box and 4 ways to fill second box.

  $\boxed{5}\boxed{4}\boxed{}\boxed{}\boxed{}$

For the third box 4 ways are there to fill it, fourth box has 3 ways and fifth box has 2 options.

  $\boxed{5}\boxed{4}\boxed{4}\boxed{3}\boxed{2}$

Therefore, the possible ways are $5\cdot 4\cdot 4\cdot 3\cdot 2=480$ .

Thus, the number of different 5-digit security codes when repetition of digits is allowed are $1600$ and when repetition of digits is not allowed are 480.