Chapter 8.5, Problem 17E

(a).

To determine

To calculate: the number of three digit formations for specific condition.

Answer to Problem 17E

The number of possible three digit numbers without leading zero is $\underset{\_}{900}$ .

Explanation of Solution

Given information:

Need to determine number of three digits which can be formed where leading digit is not zero.

Formula used:

Fundamental count principle:

If there are three events say $E_1,E_2$ and $E_3$

First event $E_1$ can occur in $w_1$ different ways, after $E_1$ has occurred, $E_2$ can occur in $w_2$ different ways and after $E_1$ and $E_2$ occurred, $E_3$ can occur in $w_3$ different ways, than three events can occurred in $w_1\times w_2\times w_3$ ways.

Calculation:

Need to form three digits number where leading number cannot be zero.

Lets say first event be $E_1$ = selection of first digit

Second event $E_2$ = selection of second digit

Third event $E_3$ = selection of third digit

As first digit cannot have 0 , so first digit can have number from 1 to 9. So there are 9 ways in which $E_1$ can occurs that is number of ways $w_1=9$ .

As repetition is allowed, second digit and third digit cannot have number from 0 to 9 that means there are 10 ways in which $E_1$ and $E_2$ can occurred.

So $w_2=w_3=10$ .

So using principle of fundamental counting, total number of 3 digits, which can be formed with no 0 at leading place $=w_1\times w_2\times w_3$

  $=9\times 10\times 10$

  $=900$

Hence number of possible three digit numbers without leading zero is $\underset{\_}{900}$ .

(b).

To determine

To calculate: the number of three digit formations for specific condition.

Answer to Problem 17E

The number of possible three digit numbers without leading zero and no repetition

is $\underset{\_}{648}$ .

Explanation of Solution

Given information:

Need to determine number of three digits which can be formed where leading digit is not zero and no repetition of number is allowd.

Formula used:

Fundamental count principle:

If there are three events say $E_1,E_2$ and $E_3$

First event $E_1$ can occur in $w_1$ different ways, after $E_1$ has occurred, $E_2$ can occur in $w_2$ different ways and after $E_1$ and $E_2$ occurred, $E_3$ can occur in $w_3$ different ways, than three events can occurred in $w_1\times w_2\times w_3$ ways.

Calculation:

Need to form three digits number where leading number cannot be zero.

Lets say first event be $E_1$ = selection of first digit

Second event $E_2$ = selection of second digit

Third event $E_3$ = selection of third digit

As first digit cannot have 0 , so first digit can have number from 1 to 9. So there are 9 ways in which $E_1$ can occurs that is number of ways $w_1=9$ .

As repetition is not allowed, second digit can have number between 0 to 9 except the one which got selected in $E_1$ as first digit, so number of ways in which $E_2$ can occurred that is $w_2=9$

As repetition is not allowed, third digit can have number between 0 to 9 except the two numbers which got selected in $E_1$ and $E_2$ as first and second digits , so number of ways in which $E_3$ can occurred that is $w_3=8$

So using principle of fundamental counting, total number of 3 digits, which can be formed with no 0 at leading place and no repetition $=w_1\times w_2\times w_3$

  $=9\times 9\times 8$

  $=81\times 8$

  $=648$

Hence number of possible three digit numbers without leading zero and no repetition is $\underset{\_}{648}$ .