This is part of the “Memorize or Melt” series, for more information go here if you haven’t read the introductory post on this topic.

FOIL stands for First Outer, Inner Last. FOIL is a mnemonic device that is supposed to help students expand parenthetical statements. For example, to expand \((a + b)(c + d)\) via FOIL, the procedure is to first handle the “outer” term and then handle the “inner” term. The outer term is \(a\) and the inner term is \(b\). Thus, FOIL dictates that first \(a\cdot c + a\cdot d\) then \(b\cdot c + b\cdot d\). Combine these to give \(a\cdot c + a\cdot d + b\cdot c + b\cdot d\). This is a good clean method. Reliable, repeatable, safe, and programmable as a recursive method. However, since this is the “Memorize or Melt” series, it means that there are problems with FOIL. Technically though, the problem isn’t FOIL itself, it is how FOIL is taught.

Students are taught FOIL exactly as explained above. The problem is that is all they are taught. FOIL taught in this way is very limiting and soulless. When students are given a slight variation on this expansion, say, \((a+b+c)(x+y+z)\), they are lost. The presence of a third variable in the parentheticals confuses students who are lead to believe that there is an inner and outer only. But now there is a “middle”! What to do then? Introduce FOSMIL? First Outer, Second Middle, Inner Last? Again what would that even mean?

To use the FOIL method in a case like this, one would rewrite the problem possibly like this: \(((a+b) + c)((x+y) + z)\) where the outer term is \((a+b)\) and the inner term is \(c\). This provides a recursive method consistent with FOIL for expanding the expression. In other words, encapsulating certain terms in parentheses, FOIL would give \((a+b)(x+y) + (a+b)z + c(x+y) + cz\). Then the next step would be use FOIL again on \((a+b)(x+y)\) and use the distributive property on the middle two terms. There is absolutely nothing wrong with the mechanics or the general notion of FOIL in this way.

However, what is the goal here? Is it to show a vacuous process?

This entire way of teaching is evidence that the intent of such methods was to allow students to process math as if problems were coming down an assembly line — nice, tidy, and of the same form. This is terrible!! This type of pedagogy is antiquated and simply wrong. All combinatorial meaning has been sucked out by teaching FOIL. The mechanics are fine, if they were taught correctly (and they are not taught correctly), but what about some meaning?

Let’s try this again. Here is a simple problem. Suppose we have two, three-sided dice with each die having the numbers 1, 2, and 3 written (one number per face). Then, what is the distribution of the sum of the two dice? There is more than one way to answer this question. One way is the classical, tabular method shown below. The first row gives the three possible die values for one of the dice and the first column does the same for the other die. The other nine values are the individual sums for the possible outcomes. Counting the occurrence of each sum, it is easy to generate the distribution of the sum.

The distribution of the sum of two, three-sided dice

Die One + Die Two	1	2	3
1	2	3	4
2	3	4	5
3	4	5	6

From this, it is clear to see that a sum of two occurs one in nine times, a sum of three occurs two in nine times, and so forth.

Another way to solve this problem is to set up the following generating function. (For an excellent resource on generating functions, check out Professor Herbert Wilf’s book generatingfunctionology.) Have students construct the table above and do the necessary counting to build the distribution. Then have students expand, term by term and without calling it FOIL, $$(x^{1} + x^{2} + x^{3})(x^{1} + x^{2} + x^{3})$$ This will resolve to $$x^{2} + 2x^{3} + 3x^{4} + 2x^{5} + x^{6}$$

Notice that the exponents are the possible sums and the coefficients give the respective frequency for the sum! This is a wonderful example to show students that this is not just a pointless exercise of following a prefabricated set of steps. The very act of multiplying term by term is generating the distribution of the sum of two, three-sided dice.

An instructor can really begin to play and vary the theme. Suppose the dice were not numbered 1, 2, and 3, but instead one die were numbered -1, 0, and 1 and the other 1, 1, and 2. What then? It would be the “same” set up with the same mechanics but a different distribution. To solve this would require resolving \((x^{-1} + x^{0} + x^{1})(x^{1} + x^{1} + x^{2})\) (to be explicit).

Hopefully, the creative juices can begin to flow. What if instead of two, three-sided dice there were one, three-sided die and one, four-sided die? What about having three, three-sided die? What would the table look like? What would the generating function look like? How many multiplications would there have to be? What kind of dice game can be made? What about coin flips? What about a deck of cards? What if there were a game that used one, six-sided die and one coin? What if the dice were not equally weighted?

Now, students are doing the mechanics as they should, but there is some purpose and there is some creativity!

Here is how the sum of three, three-sided dice would resolve. First as a table, then via the generating function.

The distribution of the sum of three, three-sided dice

Die One + Die Two	1	2	3	+ Die Three	1	2	3
1	2	3	4		3	4	5
					4	5	6
					5	6	7
2	3	4	5		4	5	6
					5	6	7
					6	7	8
3	4	5	6		5	6	7
					6	7	8
					7	8	9

The generating function gives $$(x^{1}+x^{2}+x^{3})^{3} = x^{3} + 3x^{4} + 6x^{5} + 7x^{6} + 6x^{7}+ 3x^{8} + x^{9}$$

Then, when students are eventually introduced to the binomial theorem (and perhaps the multinomial theorem) (maybe now is a good time to give them a little preview?), they are not lost again in a maze of procedures.

I encourage instructors to judiciously leave some of the old ways in the past. FOIL had its time, but there are so many other, better ways to explain the mechanics in a lasting, meaningful way that does not rely on acronyms and repetitive exercises to build muscle memory.