An ordered arrangement of objects is called a \textit{permutation}. An unordered selection\footnotemark{} of objects is called a \textit{combination}. All the following problems involve permutations.
\footnotetext{A \say{combination lock} cares about the order of its digits, so its name is inaccurate. Such an object is actually a \textit{permutation} lock!}
\hint{When you see a problem that's as big as this one, it's often wise to try and understand a simpler case first. Look at \ref{ABCDE} again, and try to create a general strategy.}
Consider the sequence RRRRGGG. There are $4!$ ways to rearrange the red balls, and $3!$ ways to rearrange the green balls. This is true for any sequence.
\hint{We're counting \textit{permutations}, so the order of items in the first group matters.}
\begin{solution}
$\npk{n}{k}=\frac{n!}{(n-k)!}$
There are $n!$ possible arrangements of $n$ objects. However, since the order of the elements not chosen does not matter, we'll end up with $(n-k)!$ redundant orderings of each.
Here, we care about \textit{which} items we choose, but not \textit{how} we choose them. We'll make a function $\nck{n}{k}$, or \textit{``n choose k''}. This will tell us how many different ways we can choose $k$ items from a set of $n$.
\problem{}
Find an expression for $\nck{n}{k}$ by modifying your definition of $\npk{n}{k}$.
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Usually, $\nck{n}{k}$ is written as $\binom{n}{k}$. This is also called the \textit{binomial coefficient}.
Now, derive the \textit{multinomial coefficient}$\binom{n}{k_1,k_2,...,k_m}$. \par
\vspace{1mm}
The multinomial coefficient tells us how many distinct ways we can choose $n$ objects from a set which has $m$ classes, and where each class $i$ contains $k_i$ identical objects. \par
How many ways can you put 19 identical balls into 6 bins, so that no bin is empty?
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\problem{}
Given an exam with 4 problems, how many ways are there to assign positive point values to each problem so that the exam contains a total of 100 points?
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\problem{}
How many ways can we split the number 2016 into a sum of positive integers?
A staircase must be built up a wall. It will start 4.5 meters away from the wall, which is 1.5 meters tall. The height of each step is exactly 30 centimeters. The width of each step must be an integer multiple of 50 centimeters. In how many ways can the staircase be constructed?
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\section{Bonus problems}
\problem{}
A toy consists of a ring with 3 red beads and 7 blue beads on it. If two configurations of beads differ only by rotations and reflections, they are considered the same toy. How many different toys are there?
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\problem{}
At a math circle meeting, 10 students are given 10 problems. Any two students solved a different number of problems, and every problem is solved by the same number of students. Yan solved problems 1 through 5, but did not solve problems 6 through 9. Did he solve problem 10?
