Get Expert Help with the Fundamental Counting Principle

1 octopus has 8 legs. How many legs does 3 octopuses have?
A) 16
B 24
C) 32
D) 14

What are the applications of Archimedes' principle ? AIt is used in designing of ships and submarines. BIt is used in lactometers to determine the purity of milk. CIt is used in hydrometers to determine density of fluids. DIt is used in hydraulic lifts.

Determine the number of different ways that the letters of "success" can be arranged

How many license plates can be made consisting of 2 letters followed by 3 digits (using the fundamental counting principle to solve)? What I know:
1. 26 letters in alphabet, so that means ${\displaystyle 2\times 26}$
2. 10 digits possible (0-9), so that means ${\displaystyle 3\times 10}$
3. FC principle says given m and n options gets you ${\displaystyle m\times n}$ varieties...
... However, the answer key says "676,000" when I got 1560...

Different 4 math books , 6 different physics books , 2 different chemistry books are to be arranged on a shelf. how many arrangement is possible if the books in particular subject must stand together ?

Of a group of 50 students, 20 are freshmen, 10 are sophomores, 15 are juniors, and 5 are seniors. 5 members must be chosen. How many different committees can there be if there must be exactly one senior and exactly two freshmen on the committee?

How do you use the Counting Principle to find the probability of choosing the 5 winning lottery numbers when the numbers are chosen at random from 0 to 9?

A bag contains 8 white marbles and 7 blue marbles. Find the probability of selecting 2 white marbles and 3 blue marbles: (hint: use combinations and fundamental counting principle)
It doesn't say it, but I'm sure there are no replacements. I know it's a combination, but I don't know how 3C2 (enteblack it equals 3) would be the answer because just "3" doesn't sound right.

A piece of string ${\displaystyle 25\frac{1}{2}}$ inches long will be cut into ${\displaystyle \frac{3}{4}}$-inch pieces. How many pieces will there be?

There are ${\displaystyle \frac{m}{gcd(m,x)}}$ distinct elements in the set $\{ax(\mathrm{mod}m):a\in <!--\; \in \; -->\{0,...,m-<!--\; -\; -->1\}\}$
I have only known these by using a computer to generate the number of distinct elements. But I am not sure how to prove this conjecture.
And is there any way that we can connect this problem to Euler's phi function so that we can simply use properties of $\mathrm{\varphi <!--\; \varphi \; -->}$ function to prove it?
And can we also use some counting principle here to give an exact answer?

Find the number of Sylow 2-subgroups of the special linear group of order 2 on $\mathbb{Z}$ (modulo 3). I think it will be 1. But I failed to prove it using the counting principle. It has 4 sylow 3-subgroups.

How many numbers $n<100$ are not divisible by a square of any integer greater than 1?
Working through the above counting problem. I got 48 using the Inclusion-Exclusion Principle, do you agree?

In how many ways can 7 people be seated in a row o chairs if Jane and Joe must sit next to each other?

The next three batters on a baseball team have hit percentages of 0.325, 0.250, and 0.275, respectively. What is the probability that the first and third batters will both get a hit, while the second batter does not?

I tried the Fundamental Counting principle approach: There are 26 black cards to chose from for the first card. That leaves $26\times <!--\; \times \; -->25\times <!--\; \times \; -->24\times <!--\; \times \; -->23$ left for the other four cards. So $\frac{26\times <!--\; \times \; -->26\times <!--\; \times \; -->25\times <!--\; \times \; -->24\times <!--\; \times \; -->23}{5!}$ (because we can arrange 5 cards in $5!$ ways). I get 77740. Apparently the answer is: 454480.

I'd appreciate help clearing up my misunderstanding. I'd also like to see the combinations approach. Thanks!

I was doing a seemingly trivial question, and I though it was a simple application of the counting theorem but it turns out it doesn't work. Here's the question

From a deck of 52 cards, how many ways are there to arrange a hand of 5 cards such that all 4 kings are in the hand (order doesn't matter) (the last card can be any non-king)

Now here's my thought process as an application of the counting principle:
$\frac{4\times <!--\; \times \; -->3\times <!--\; \times \; -->2\times <!--\; \times \; -->1\times <!--\; \times \; -->48}{5!}$
As we have 4! ways of placing the kings and then the last card can be from 48 other cards. Then we divide by 5! to remove the order. Unfortunately, this produces a non-integer so I was very sad indeed. However, it logically seems like it should work as it follows what I think is valid logic. Could someone explain how to get the correct answer (48) and also more importantly, why my logic was incorrect?

In a computer game, when choosing a character there are 5 choices for hair style, 3 choices for eye color, and 7 choices for outfits. How many different characters are possible if you were to choose 1 hair style, 1 eye color, and 1 outfit?

Let X be the set of 10-digit numbers that do not contain all digits 0 through 9 in their decimal representation and do not begin with 0. How many elements does the set X have?
I think this should be done by counting principle. So I have 10 spots, and 10 options for each spot. However, the first number can't be 0, so the first spot has 9 options.
9 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 9
I think the 10th spot only have 9 options since I can't repeat one number (the representation doesn't have all the digits).

How do you use the Counting Principle to find the probability of rolling a 4 on each of 4 number cubes?

There are two fundamental principles of counting; Fundamental principle of addition and fundamental principle of multiplication.

I often got confused applying them. I know that if there are two jobs, say $m$ and $n$, such that they can be performed independently in $m$ and $n$ ways respectively, then either of the two jobs can be performed in $m+n$ ways and when two jobs are performed in succession, they can be performed in $m\times <!--\; \times \; -->n$ ways.

My question is how to identify whether jobs are independent or in succession?

Is there any simple way to identify this? Are there any keywords?