Multiplication is a wonderful little operation. Depending on the context, it can

And today weâ€™ll see yet another use: listing combinations.

Revisiting multiplication has a few uses:

• It demystifies other parts of math. The binomial theorem, Boolean algebra (used in computer circuits) and even parts of calculus become easier with a new interpretation of â€śmultiplicationâ€ť.
• It keeps our brain fresh.Math gives us models to work with, and itâ€™s good to see how one model can have many uses. Even a wrench can drive nails, once you understand the true nature of â€śbeing a hammerâ€ť (very Zen, eh?).

The long multiplication you learned in elementary school is quite useful: we can find the possibilities of several coin flips, for example. Letâ€™s take a look.

## Youâ€™ve Been Making Combinations All Along

How would you find 12 Ă— 34? Itâ€™s ok, you can do it on paper:

â€śWell, letâ€™s seeâ€¦ 4 times 12 is 48. 3 times 12 is 36â€¦ but itâ€™s shifted over one place, so itâ€™s 360. Add 48 and 360 and you getâ€¦ uhâ€¦ carry the 1â€¦ 408. Phew.â€ť

Not bad. But instead of doing 12 Ă— 34 all at once, break it into steps:

Whatâ€™s happening? Well, 4 Ă— 12 is actually â€ś4 x (10 + 2)â€ť or â€ś40 + 8â€ł, right? We can view that first step (blue) as two separate multiplications (4Ă—10 and 4Ă—2).

Weâ€™re so used to combining and carrying that we merge the steps, but theyâ€™re there. (For example, 4 Ă— 17 = 4 x (10 + 7) = 40 + 28 = 68, but we usually donâ€™t separate it like that.)

Similarly, the red step of â€ś3 Ă— 12â€ł is really â€ś30 Ă— 12â€ł â€” the 3 is in the tens column, after all. We get â€ś30 x (10 + 2)â€ť or â€ś300 + 60â€ł. Again, we can split the number into two parts.

What does this have to do with combinations? Hang in there, youâ€™ll see soon enough.

## Curses, Foiled Again

Take a closer look at what happened: 12 Ă— 34 is really (10 + 2) x (30 + 4) = 300 + 40 + 60 + 8. This breakdown looks suspiciously like algebra equation (a + b) * (c + d):

And yes, thatâ€™s whatâ€™s happening! In both cases weâ€™re multiplying one â€śgroupâ€ť by another. We take each item in the red group (10 and 2) and combine it with every element of the blue group (30 and 4). We donâ€™t mix red items with each other, and we donâ€™t mix blue items with each other.

This combination technique is often called FOIL (first-inside-outside-last), and gives headaches to kids. But itâ€™s not a magical operation! Itâ€™s just laying things out in a grid. FOIL is already built into the way we multiply!

When doing long multiplication, we â€śknowâ€ť weâ€™re not supposed to multiply across: you donâ€™t do 1 Ă— 2, because theyâ€™re in the same row. Similarly, you donâ€™t do a x b, because theyâ€™re in the same parenthesis. We only multiply â€śup and downâ€ť â€” that is, we need an item from the top row (1 or 2, a or b) and an item from the bottom row (3 or 4, c or d).

Everyday multiplication (aka FOIL) gives us a way to crank out combinations of two groups: one from group A, another from group B. And sometimes itâ€™s nice having all the possibilities as an equation.

## Examples Make It Click

Letâ€™s try an example. Suppose we want to find every combination of flipping a coin twice. Thereâ€™s a few ways to do it, like using a grid or decision tree:

Thatâ€™s fine, but letâ€™s be different. We can turn the question into an equation using the following rules:

• multiplication = AND. We have a first toss AND a second toss: (h+t) * (h+t)

Wow! How does this work?

Well, we really just want to crank out combinations, just like doing (a+b) * (c+d) = ac + bc + ad + bd. Looking carefully, this format means we pick a OR b, and combine it with one of c OR d.

When we see an addition (a+b), we know it means we must choose one variable: this OR that. When we see a multiplication (group1 * group2), we know it means we take one item from each: this AND that.

The shortcuts â€śAND = multiplyâ€ť & â€śOR = addâ€ť are simply another way to describe the relationship inside the equation. (Be careful; when we say three hundred and four, most people think 304, which is right too. This AND/OR trick works in the context of describing combinations).

So, when allâ€™s said and done, we can turn the sentence â€ś(heads OR tails) AND (heads OR tails)â€ť into:

$\displaystyle{(h + t)\cdot(h + t)}$

And just for kicks, we can multiply it out:

$\displaystyle{(h + t) \cdot (h + t) = h^2 + th + ht + t^2 = h^2 + 2ht + t^2}$

The result â€śh2 + 2ht + t2â€ť shows us every possibility, just like the grids and decision trees. And the size (coefficient) of each combination shows the number of ways it can happen:

• 2ht: Thereâ€™s two ways to get a head and tails (ht, th)
• t^2: Thereâ€™s one way to get two tails (tt)

Neato. The sum of the coefficients is 1 + 2 + 1 = 4, the total number of possibilities. The chance of getting exactly one heads and one tails is 2/4 = 50%. We figured this out without a tree or grid â€” regular multiplication does the trick!

## Grids? Trees? I Figured That Out In My Head.

Ok, hotshot, letâ€™s expand the scope. How many ways can we get exactly 2 heads and 2 tails with 4 coin flips? Whatâ€™s the chance of getting 3 or more heads?

Our sentence becomes: â€ś(heads OR tails) AND (h OR t) AND (h OR t) AND (h OR t)â€ť

$\displaystyle{(h+t)^4 = h^4 + 4h^3t + 6h^2t^2 + 4ht^3 + t^4}$

Looking at the result (it looks hard but there are shortcuts), there are 6 ways to get 2 heads and 2 tails. Thereâ€™s 1 + 4 + 6 + 4 + 1 = 16 possibilities, so thereâ€™s only a 6/16 = 37.5% chance of having a â€śbalancedâ€ť result after 4 coin flips. (Itâ€™s a bit surprising that itâ€™s much more likely to be uneven than even).

And how many ways can we get 3 or more heads? Well, that means any components with h3 or h4: 4 + 1 = 5. So we have 5/16 = 31.25% chance of 3 or more heads.

Sometimes equations are better than grids and trees â€” look at how much info we crammed into a single line! Formulas work great when you have a calculator or computer handy.

But most of all, we have another tool in our box: we can write possibilities as equations, and use multiplication to find combinations.

## Where Next?

Thereâ€™s a few areas of math that benefit from seeing multiplication in this way:

• Binomial Theorem. This scary-sounding theorem relates (h+t)^n to the coefficients. If youâ€™re clever, you realize you can use combinations and permutations to figure out the exponents rather than having to multiply out the whole equation. This is what the binomial theorem does. Weâ€™ll cover more later â€” this theorem shows up in a lot of places, including calculus.

• Boolean Algebra. Computer geeks love converting conditions like OR and AND into mathematical statements. This type of AND/OR logic is used when designing computer circuits, and expressing possibilities with equations (not diagrams) is very useful. The fancy name of this technique is Boolean Algebra, which weâ€™ll save for a rainy day as well.

• Calculus. Calculus gets a double bonus from this interpretation. First, the binomial theorem makes working with equations like x^n much easier. Second, one view of calculus is an â€śexpansionâ€ť of multiplication. Today we got practice thinking that multiplication means a lot more than â€śrepeated additionâ€ť. (â€ś12 Ă— 34â€ł means 12 groups of 34, right?)

• More advanced combinations. Letâ€™s say you have 3 guests (Alice, Bob, and Charlie) and they are bringing soda, ice cream, or yogurt. Someone knocks at the door â€” what are the possibilities? (a + b + c) * (s + i + y). The equation has it all there.

So you can teach an old dog like multiplication new tricks after all. Well, the tricks have always been there â€” itâ€™s like discovering Fido has been barking poetry in morse code all this time.

And come to think of it, maybe weâ€™re the animal that learned a new trick. The poetry was there, staring us in the face and we just didnâ€™t recognize it (12 Ă— 34 is based on combinations!). I know I had some forehead-slapping moments after seeing how similar combinations and regular multiplication really were.

Happy math.