Understanding the concept of multiplication

Question

Since multiplication is such a basic algorithm in math we rarely stop to think about what it really is about, so please, help me understand?

When we multiply two numbers we are basically increasing the number of units of one number by the other. When we calculate factorials for example, why is it we multiply one number by the other?

If I want to know how many ways $5$ people can stand in line, why would I need to multiply $5 \times 4 \times 3 \times 2 \times 1$ if I just want the total number, which seems like I should add them all? In other words why is it I need the number $5$ increased by $4$ units (multiplication), then $3$ units, then $2$, etc?

What does it mean to multiply?!

Answer 1

The simplest sort of multiplication works like this: $$ \begin{array}{ccccc} \bullet & \bullet & \bullet & \bullet & \bullet \\ \bullet & \bullet & \bullet & \bullet & \bullet \\ \bullet & \bullet & \bullet & \bullet & \bullet \end{array} $$ You have three horizontal rows each with five things, so $3\times5 = 5+5+5=15$ things,

or you have five vertical columns each with three things, so $5\times 3= 3+3+3+3+3=15$ things.

Now consider your five people standing in a queue. Their names are A, B, C, D, E.

Who is first in the queue? It's one of the five. Who is second in the queue? It's one of the other four: $$ \begin{array}{ccccc} A\text{ is first} & B \text{ is first} & C\text{ is first} & D\text{is first} & E\text{ is first} \\ \hline AB & BA & CA & DA & EA \\ AC & BC & CB & DB & EB \\ AD & BD & CD & DC & EC \\ AE & BE & CE & DE & ED \end{array} $$ So for each possible answer to "who is first", there are four possible answers to "who is second", and thus $5\times 4$ possibilities.

Now as "who is third?". It's one of the three remaining possibilities after the first two. If the first two are $AB$, then you can have $ABC$ or $ABD$ or $ABE$. If the first two are $EC$, then you can have $ECA$ or $ECB$ or $ECD$. And so on. For every one of the $20$ possibilities in the table above, there are three possibilities, so you get a list three times as long as the table above, thus $5\times4\times3=60$.

And so on.

Answer 2

Multiply just means you want a certain amount repeated a certain number of times. It's like if you have 3 different pants(A,B,C) and 3 different shirts(1,2,3) and want to know how many outfits you can create.

You have to take the set of 3 pants and pair them with a different shirt 3 times.(1A, 1B, 1C) and (2A, 2B, 2C), and (3A, 3B, 3C). We call this $3\times 3$.

So, expanding on this, to the 5 people standing in line, for every way a person can be placed in the first position (5), there are 4 ways a person can be placed in the second position. So there are $5\times 4$ ways to fill in the first two spots. And for every one of those 20 ways to fill the first two spots, there are 3 ways to fill the third spot. $5\times 4\times 3$.

ETA: This is a very simplified explanation only dealing with whole numbers.

Alternately to all this is thinking of multiplying as a grid area. Google the subject for much better drawings and explanations.