Question

(a) How many different license plates involving three letters and two digits are there if the three letters appear together either at the beginning or end of the license? (b) How many license plates involving one, two, or three letters and one, two, or three digits are there if the letters must appear in a consecutive grouping?

Answer 1

Answer:

a) 3,515,200 license plates

b) 78,960,960 license plates

Step-by-step explanation:

a) The letters appear together and the numbers appear together too, taking the 3 letters as one big letter, and the 2 numbers as one digit, they can each occupy either of the 2 spots; at the start of at the end.

Then, ten digits can each take a spot of the 2 digits number and 26 alphabets can each take a spot of the three letters.

2 × [26 × 26 × 26] × [10 × 10] = 3,515,200

b) Since the letters will appear together as a group, we can work with them as one big letter.

- If there are 3 digits, then there are 4 spots for where the big letter goes (no matter if there are 1, 2 or 3 letters).

- If there are 2 digits, then there are 3 spots for where the big letter goes (no matter if there are 1, 2 or 3 letters).

- If there is only 1 digit, then there are 2 spots for where the big letter goes (no matter if there are 1, 2 or 3 letters).

Thus, the answer is:

(number of license plates with one digit) + (number of license plates with two digits) + (number of license plates with three digits)

Note that there are 10 numbers (0 to 9) that can fill the 1 digit, 2 digits and 3 digits spots each.

And there are 26 alphabets that can each fill out the 1 letter, 2 letters and 3 letters spots too.

[2 × 10 × (26 + 26² + 26³)] + [3 × 10² × (26 + 26² + 26³)] + [4 × 10³ × (26 + 26² + 26³)] = 78960960

Hope this Helps!!!

Answer 2

The number of different license plates in both cases are 1757600 and 7263360, respectively

1. The number of license plates

There are 26 upper case alphabets and 10 digits.

Since the alphabets appear together at the beginning and the digits appear together at the end, then:

• The first character can be selected from any of the 26 alphabets
• The second character can be selected from any of the 26 alphabets
• The third character can be selected from any of the 26 alphabets
• The first character can be selected from any of the 10 digits
• The second character can be selected from any of the 10 digits

So, the number of different license plates is:

$n = 26 * 26 * 26 * 10 * 10$

$n = 1757600$

2. The number of license plates

For one letter:

• The first character can be selected from any of the 26 alphabets
• The other four characters can be selected from any of the 10 digits

For two letters:

• The first and the second characters can be selected from any of the 26 alphabets
• The other three characters can be selected from any of the 10 digits

For three letters:

• The first, second and the third characters can be selected from any of the 26 alphabets
• The other two characters can be selected from any of the 10 digits

So, the number of different license plates is

$n = 26 * 10^4 + 26^2 * 10^3 + 26^3 * 10^2 + 26^4 * 10^1$

$n = 7263360$

Hence, the number of different license plates is 7263360

Read more about permutation and combination at:

https://brainly.com/question/12468032