Chapter 9.6, Problem 24E

a.

To determine

Find four-digit number.

Answer to Problem 24E

  $9000$

Explanation of Solution

Calculation:

The fundamental counting principal, if two events $E_{1}and{E}_2$ can occur in $m_1$ and $m_2$ different ways respectively, and $E_2$ occurs after $E_1$ has occurred, the number of ways that two events can occur is $m_1\cdot m_2$

Consider that $4$ digit numbers to be formed under given conditions,

The leading digit cannot be zero, since the first digit cannot be zero, we can choose the first digit is $9$ . , Since, repetition is allowed, the number of ways in which we can choose the rest og the $3$ digits is $10$ each.

The fundamental principal of counting the number of such $4$ digit is given by.

  $\begin{array}{l}{m}_1\cdot m_2\mathrm{...}{m}_4=9\cdot 10\cdot 10\cdot 10\\ =9000\end{array}$

Hence, the $4$ digit numbers without the leading zero is $9000$ .

b.

To determine

Find four-digit number.

Answer to Problem 24E

  $4536$

Explanation of Solution

Calculation:

The leading digit cannot be zero and no repetition of digits is allowed. The third and fourth digit can be chosen in $8and7$ ways respectively, therefore

The fundamental principal of counting the number of such $4$ digit is given by.

  $\begin{array}{l}{m}_1\cdot m_2\mathrm{...}{m}_4=9\cdot 9\cdot 8\cdot 7\\ =4536\end{array}$

Hence, the $4$ digit numbers without the leading zero and no repetition of digits is $4536$ .

c.

To determine

Find four-digit number.

Answer to Problem 24E

  $4000$

Explanation of Solution

Calculation:

The leading digit cannot be zero and number must be less than $5000$ .since the first digit cannot be zero

We can choose first digit $4$ since repetition is allowed, the number of ways in which we can choose the rest of $3$ digits is $10$ each.

The fundamental principal of counting the number of such $4$ digit is given by.

  $\begin{array}{l}{m}_1\cdot m_2\mathrm{...}{m}_4=4\cdot 10\cdot 10\cdot 10\\ =\boxed{4000}\end{array}$

Hence, the $4$ digit numbers without the leading zero and must be less than $5000$ .

is $4000$ .

d.

To determine

Find four-digit number.

Answer to Problem 24E

  $4500$

Explanation of Solution

Calculation:

The leading digit cannot be zero and the number must be even since the first digit cannot be zero

We can choose first digit $9$ since repetition is allowed, the number of ways in which we can choose the second and third digits is $10$ each.

The fundamental principal of counting the number of such $4$ digit is given by.

  $\begin{array}{l}{m}_1\cdot m_2\mathrm{...}{m}_4=9\cdot 10\cdot 10\cdot 5\\ =\boxed{4500}\end{array}$

Hence, the $4$ digit numbers without the leading zero the number must be even is $4500$ .