Combinatorics: Examples, Equations, and Practice Problems

Let it be a tetrahedron with the numbers 1,2,3 and 4 on its faces.The tetrahedron is launch 3 times. Each time, the number that stays face down is registered.
1 )In total how many possible ways there are to registered the 3 launches?
As there are 4 numbers to 3 launches(positions), the order matters and each number can repeats itself. I used a permutation with replacement: $4^3=64$
3 )How many possible ways there are to the number 1 appears only 1 time face down?
There are 3 ways for number 1 can be put on the 3 launches. For the 2 left there is {2,3,4}.So I made a permutation with replacement: $3\cdot <!--\; \cdot \; -->3^2=27$
4)How many possible ways there are to the number 1 appears exactly 2 times face down?
First I made a combination: C(3,2) to find the number of ways that the pair of 1's can be put in the 3 launchs.Then I multiplied by 3, that is {2,3,4} : $C(3,2)\cdot <!--\; \cdot \; -->3=9$
Is this correct?

How many ways can a rectangle be partitioned by either vertical or horizontal lines into n sub-rectangles? At first I thought it would be:
f(n) = 4f(n-1) - 2f(n-2)
where f(0) = 1
and f(1) = 1
but the recurrence relation only counts the cases in which at least one side (either top, bottom, left or right of the original rectangle) is not split into sub-rectangles.

The Fibonacci sequence is very well known, and is often explained with a story about how many rabbits there are after $n$ generations if they each produce a new pair every generation. Is there any other reason you would care about the Fibonacci sequence?

Fibonacci sequence can be described via the Binet's formula.
However, I was wondering if there was a similar formula for $n!$.
Is this possible? If not, why not?

Each person from a group of 3 people can choose his dish from a menu of 5 options. Knowing that each person eats only 1 dish what are the number of different orders the waiter can ask the chef?

Let $X$ be a set, $|X|=n$ and $G$ be a group with a $2-<!--\; -\; -->$−transitive action on $X$. what can be said about the size of $G$?

How can I determine the size of the largest collection of $k$-element subsets of an $n$-element set such that each pair of subsets has at most $m$ elements in common?

If there are 20 people with 17 men and 3 women and they seat themselves randomly at 4 Tables (A,B,C,D) with 5 people each. With all arrangements being equally likely, what is the probability that no woman sits at table A ?

How many number of three digit numbers lying between 100 and 999 and having only two consecutive digits identical?

The only contents of a container are 10 disks that are each numbered with a different positive integer from 1 through 10, inclusive. If 4 disks are to be selected one after the other, with each disk selected at random and without replacement, what is the probability that the range of the numbers on the disks selected is 7?

n total antennas of which m are defective, confused by reasoning?
Basically there are a total of n antennas, of which m are defective. In how many ways can these n antennas be arranged so that no two defective antennas are side-by-side with each other?
I understand the reasoning presented in the answer to this problem in the link above. But consider the case where n = 9 and m = 7. In this case, can we say that there is no solution, because the given $(\genfrac{}{}{0ex}{}{n-<!--\; -\; -->m+1}{m})$ doesn't work?

Given an $n>0$ is it possible to partition the set $\mathcal{P}=\{1,2,\cdots <!--\; \cdots \; -->,2n\}$ into $n$ pairs $(a_i,b_i)$ such that $a_i+b_i$ is a prime?

A motel has ten rooms, all located on the same side of a single corridor and numbered 1 to 10 in numerical order. The motel always randomly allocates rooms to its guests. There are no other guests besides those mentioned.
a) Friends Molly and Polly have been allocated two separate rooms at the motel. What is the likely number of rooms between their rooms?
b) Molly believes there is a greater than 1/3 chance that at most one room will separate them, but Polly disagrees. Who is right? Explain why.
c) On another occasion, Molly, Polly and a third friend, Ollie, were allocated three separate rooms. Molly believes there is a better than 1/3 chance that they are all within a block of five consecutive rooms. Ollie believes that there is exactly 1/3 chance and Polly believes there is less than 1/3 chance. Who is right? Explain why.
d) Ollie arrived after rooms were allocated to Molly and Polly. There was then a 50% chance he would be in a room adjacent to Molly or Polly or both. In how many ways could a pair of rooms have been allocated to Molly and Polly?
Here I am bit sure of how to work through (c). Do I use permutations or combinations - could you please give me a step by step solution as I am still a bit wobbly on combinatorics.
Also could someone please check my answer for (d) (30)?

Assuming seven standard dice are rolled, what is the probability their sum equals 17? Show a general approach to solving this problem analytically, using conditional probability, combinatorics, etc

Let us consider that there are G girl students and B boy students in a class, we need to arrange them in a single row, but arrangement of students should be in order to minimize the gender regularity*. Find the minimum gender regularity among all possible arrangements.

1.Find the number of ways in which a mixed double tennis game can be arranged between 10 players consisting of 6 men and 4 women.
2.Find the number of ways in which a mixed double game can be arranged from amongst nine married couples, if no husband and wife play in the same game.

The number of ways in which $mn$ different items can be divided equally int $m$ groups, each containing $n$ objects and the order of the groups is important,is: $\frac{(mn)!}{(n!{)}^m}\frac{1}{m!}\ast <!--\; \ast \; -->(m!)$

100 people seat at random on a plane with 100 seats while they also have a ticket with an assigned seat , what is the probability to seat on your assigned seat if you are the 34th to get on the plane?

A class consists of 3 boys and 6 girls willing to form 3 groups of 3 called Groups A, B, C.
How many ways are there to assign them to the groups such that exactly 1 group has no boys?

A bus follows its route through nine stations, and contains six passengers. What is the probability that no two passengers will get off at the same station?