(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?

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!!!

1 number

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.