Permutation vs Variation - To rank 3 people

Question

I am new to permutation and combination and am looking for guidance in the following example:

We have 3 people - A, B, C

How many ways are there to arrange them into Rank 1,2,3

Looking at the example, it is clear that No repetitions are allowed and that ordering is not important (in the sense - Rank 1 - A, Rank 2 - B, Rank 3 - C is the same as Rank 2 - B, Rank 3 - C, Rank 1 - A).

So as a permutation problem we have answer as 3! = 6

Where as a variation problem we have the answer as 3!/0! = 3! = 6 (but if ordering is not important then it would be 3^3 = 27)

Please can you help me understand how me decide between Permutation or variation? and whether ordering is important or not?

Answer 1

To arrange $n$ items in a row (which can be accomplished in $n!$ ways) is equivalent to picking $k$ of $n$ items to arrange in a row (which can be accomplished in $\frac{n!}{(n-k)!}$ ways) in the case that $k=n$.

The only difference between them is semantics and that the formula for "variations" that you refer to is simply for the more generalized case where we might choose to arrange only some of but not all of our items into a row.

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It helps to keep an example in the back of your mind on what exactly each of these formula count. For example, given the set $\{a,b,c,d\}$:

• the calculation $4!$ here can be in reference to the number of arrangements of all letters where order of letters matters and letters may not be repeated, every letter appearing exactly once each. Such an arrangement can be thought of as a permutation. Examples of things being counted here would be abcd, abdc, acbd, acdb, ...

• the calculation $\frac{4!}{(4-2)!}$ here can be in reference to the number of arrangements of only two of the letters where order of letters matters and letters may not be repeated, each letter appearing at most once each. In your terminology, this would be a "variation." Examples of things being counted here would be ab, ac, ad, ba, bc, ... Note how ab and ba are treated as being different.

• the calculation $\binom{4}{2}=\frac{4!}{2!(4-2)!}$ here can be in reference to the number of combinations of two of the letters where here order of letters does not matter, letters may not be repeated, each letter appearing at most once each. Perhaps more correctly described, it counts the number of subsets of size two of the set $\{a,b,c,d\}$. Examples of things being counted here would be $\{a,b\}, \{a,c\}, \{b,c\},\dots$. Note here that $\{a,b\}$ is treated as being the same as $\{b,a\}$ since these are equal sets.

Answer 2

A permutation asks how many ways there are to permute, or order, some number of elements from a larger set of elements. For instance, suppose you're organizing a photo with family members. Only $5$ of them will fit in the photo, but you have $9$ family members present. Then how many ways are there to take this photo, if you care about the order they're in? Well, imagine we're filling the $5$ photo slots from left to right. For the very first slot, we have $9$ options, from all the family members. Now for the second slot, we have $8$ options; the next we have $7$, then $6$, and the last slot has $5$ options to fill with. This means we must have $9\cdot8\cdot7\cdot6\cdot5$ ways to take the picture. In basic combinatorics, this is denoted the permutation, $P(n,k)$, and is the number of ways to permute $k$ objects from a group of $n$:

$$ P(n,k)=n\cdot(n-1)\cdot(n-2)\cdots(n-k+1)=\frac{n!}{(n-k)!} $$

A combination asks how many ways there are to combine elements from a larger group, not including order. For this, we can use our friend, the permutation. Imagine the same situation, lining $k$ of your $n$ family members up for a photo, but this time you don't care about order. There are $P(n,k)$ ways to do it including order; for each group of $k$ family members, there are $k!$ ways to put them in order. Thus, we want to know the muber of ways to choose family members in order divided by the number of ways to order them, which leaves us with simply the number of ways to choose them. This is the combination $C(n,k)$, or more often denoted $n\choose k$:

$$ C(n,k)= {n\choose k} = \frac{P(n,k)}{k!} = \frac{n!}{k!(n-k)!} $$

To return to your initial question: suppose we have people $A,B,C$, and want to arrange them into ranks $1,2,3$. Ordering is actually important here - to see this imagine the ranks are places in an Olympic race. Clearly $A→1,B→2,C→3$ is different from $C→1,A→2,B→3$, as $C$ and their team might argue very strongly! In this case, we have $3$ options for picking rank $1$. We then have $2$ options for rank $2$, and then only $1$ option for the final rank. This gives us $3\cdot2\cdot1=3!$ ways to do it; this is a permutation in disguise, as we are arranging $3$ objects from a pool of $3$, which there is $P(3,3)$ ways to do. And if we check we find that $P(3,3)=\frac{3!}{(3-3)!}=\frac{3!}{0!}=3!$ - exactly the result we were looking for!

The $3^3=27$ answer comes in a slightly different variation which involves neither permutations or combinations. This occurs when you allow replacement of the objects! Imagine you're pulling names from a hat, but every time you pull a name and read it, you put it back. How many ways are there to pull the names? Say we have $3$ names. For the first name we pull, we have $3$ possible names to pull. But since we put it back, on the next draw we also have $3$ possibilities! Thus, if we draw $k$ times, we have $3^k$ ways to pull the names. Applied to your problem, imagine a variation in which instead of ranks we were giving $A,B,$ and $C$ awards. If we can give multiple awards to each person, we have replacement. With $3$ people and $3$ awards, there would be $3^3=27$ ways to give these awards to the people.