Chapter 13.2, Problem 46AYU

Forming License Plate Numbers How many different license plate numbers can be made using 2 letters followed by 4 digits selected from the digits 0 through 9, if:

(a) Letters and digits may be repeated?

(b) Letters may be repeated, but digits may not be repeated?

(c) Neither letters nor digits may be repeated?

To determine

To find: How many different license plate numbers can be made using 2 letters followed by 4 digits selected from the digits 0 through 9, if:

a. Letters and digits may be repeated?

Answer to Problem 46AYU

a. 6760000

Explanation of Solution

Given:

Different license plate numbers can be made using 2 letters followed by 4 digits selected from the digits 0 through 9.

Formula used:

$P\left(n,r\right)=\frac{n!}{\left(n-r\right)!}$

Calculation:

Number of alphabet to be used is 26 (A-Z).

Number of digits to be used is 10 (0 through 9).

a. Letters and digits may b repeated.

Number of choices for first two positions is 26 each.

Number of choices for the 4 digits is 10 each.

Hence, total number of choices is $P(26,1)*P\left(26,1\right)*P\left(10,1\right)*P\left(10,1\right)*P\left(10,1\right)*P\left(10,1\right)=6760000$.

To determine

To find: How many different license plate numbers can be made using 2 letters followed by 4 digits selected from the digits 0 through 9, if:

b. Letters may be repeated, but digits may not be repeated?

Answer to Problem 46AYU

b. 3407040

Explanation of Solution

Given:

Different license plate numbers can be made using 2 letters followed by 4 digits selected from the digits 0 through 9.

Formula used:

$P\left(n,r\right)=\frac{n!}{\left(n-r\right)!}$

Calculation:

Number of alphabet to be used is 26 (A-Z).

Number of digits to be used is 10 (0 through 9).

b. Letters may be repeated, but digits may not be repeated?

Number of choices for first two positions is 26 each.

Since, digits cannot be repeated.

Number of choices for the 4 digits is $P\left(10,4\right)=\frac{10*9*8*7*6!}{6!}$.

Hence, total number of choices is $P\left(26,1\right)*P\left(26,1\right)*P\left(10,4\right)=26*26*10*9*8*7=3407040$.

To determine

To find: How many different license plate numbers can be made using 2 letters followed by 4 digits selected from the digits 0 through 9, if:

c. Neither letters nor digits may be repeated?

Answer to Problem 46AYU

c. 3276000

Explanation of Solution

Given:

Different license plate numbers can be made using 2 letters followed by 4 digits selected from the digits 0 through 9.

Formula used:

$P\left(n,r\right)=\frac{n!}{\left(n-r\right)!}$

Calculation:

Number of alphabet to be used is 26 (A-Z).

Number of digits to be used is 10 (0 through 9).

c. Neither letters nor digits may be repeated.

Choices for alphabets cannot be repeated. Hence,

Number of choices for first two positions is $C\left(26,2\right)=26*25$.

Since, digits cannot be repeated.

Number of choices for the 4 digits is $C\left(10,4\right)=\frac{10*9*8*7*6!}{\left(10-4\right)!}$.

Hence, total number of choices is $26*25*10*9*8*7=3276000$.