How do Permutations work?

Question

A mother has 5 distinct apples which she gives to her 8 sons so that each son receives either one apple or none. How many different ways can she do this?

Permutations are new to me, can someone help me begin to understand. I know the formula but don't know what numbers belong where.

Answer 1

Don't worry for now about the formulas. Just think through the problem. It's useful that the five apples (or oranges) are distinct.

Ask yourself how many choices Mom has for the first apple.

Once she's made that choice, ask yourself how many choices she has remaining for the second apple. So in total, how many choices does she have for just the first two apples?

Lather, rinse, repeat until you've gone through all five apples. (To check your work, I get $6720$.)

Answer 2

Factorials and permutations are methods of counting things. For example if you were asked to solve a code and you knew the numbers were $1,2,3,4$ but not their order, you could figure how many possible tries it could take you.

You reason, well any one of the four numbers could go first so there's four options there. Then since you've already used one of the numbers you reason that any of the remaining three could go in the second position. Since there were four options in the first and three in the second this means there are now 12 different ways of assigning the first two numbers. You pick any one for the first and any of the remaining three for the second and that will produce 12 different possible arrangements. Likewise, in the third position there are two remaining numbers and choosing one will increasing the total possible number of codes to 24. Finally you for the last position only have one option left so there is no new additional arrangements!

This the idea behind factorials and permutations. They are ways of expressing "numbers of things" where both order and membership matter (we care both what we are choosing and where we put it in sequence). In the above example it is simply $4!=4\times3\times2\times1=24$ different ways to arrange the numbers ($4!$ reads four factorial).

The idea behind permutations (at least in this context) is that you don't "finish" the factorial. Instead of giving apples to sons here how about you give sons to apples! The first apple can have any one of the 8 sons given to it, the second has any of the remaining 7, the third 6, fourth 5 and fifth 4. In total we then have $8\times7\times6\times5\times4= \frac{8!}{3!}=6720$ This is the number of possible ways to distribute these "unique" apples.

In more complicated situations you can think about arranging a deck of cards which has 52 different elements and expands to a mind boggling $8.0658175 \times 10^{67}$. Which interestingly implies that every time you shuffle a deck of cards, more likely than not the order of those cards has never been created before! If you'd like to learn more this website gives lots of nice introductory examples.