A shorthand for descending products, and why the empty product is set to one

0:00Fill three places from six objects and you write six times five times four. Fill five places from nine and the line runs to five factors. Use every object you have, and the line runs all the way down to one. That last case turns up so often it was given a name, and the name is a mark you have seen: a number with an exclamation mark after it.

0:27It introduces no new operation. It is a name for a product you were already writing out. Here is what it names, one line at a time. One on its own is one factorial. One times two is two factorial, and it comes to two. One times two times three is three factorial, which is six. One times two times three times four is four factorial, twenty-four. The pattern is the whole definition: multiply the whole numbers from one up to the number in front of the mark.

1:05Look at the list of factors. The factors of five factorial are one, two, three, four, five. The factors of four factorial are one, two, three, four. The same list, with five missing off the end. So five factorial is five times four factorial, and that is not a trick. It is what the two lists are. In general, a factorial is the number itself times the factorial one step below it.

1:35That single relation is what almost everything done with this symbol rests on, and it was got by nothing more than noticing how the two lists of factors sit inside each other. Now apply it over and over to the same number. Five factorial is five times four factorial. Apply it again to the four factorial: five times four times three factorial. Again: five times four times three times two factorial.

2:08And again: five times four times three times two times one factorial. Four lines. Every one of them is the same relation used once more, and every one of them comes to a hundred and twenty. Nothing was calculated: the number has been rewritten four ways, with more of its factors standing outside the symbol each time. Here is the interesting part. Nothing stops you doing it once more. One factorial is one times nought factorial. So the last line becomes five times four times three times two times one times nought factorial.

2:51And now you have to say what nought factorial is worth. Go back to the definition: the whole numbers from one up to the number in front of the mark. From one up to nought. There are no such numbers. The list of factors is empty. So this is not an arithmetic question at all. It is a question about what to write down when there is nothing to multiply.

3:17You could argue for zero, on the grounds that there is nothing there. You could argue for one. You could refuse to answer. So let us not argue. Let us try each and see which survives. Four candidates: nought, one, two, or a flat refusal to say. And one test they all have to pass: the relation. A factorial is the number times the factorial below it. Score each candidate at every value of n from one to nine.

3:50Start high. At n equal to five it reads a hundred and twenty equals five times twenty-four, which is true, and no candidate was consulted to get there. The same at four, at three, at two: the relation never reaches down that far, so all four agree everywhere above one. There is exactly one place any of them is tested, and it is n equal to one. There the relation reads: one factorial equals one times nought factorial.

4:24One factorial is one. So the candidate has to satisfy one equals one times itself. Nought fails, since that would make one factorial come out as nothing. Two fails, since that would make it come out as two. Refusing to answer does not make the relation false, but it does not carry it either. It leaves the last step unspoken. One passes. Out of nine values of n, the counts are eight, nine, eight and eight, so exactly one candidate carries the relation the whole way down.

5:03Put the cascade back up and work out all five lines under each candidate in turn. The first four lines never touch nought factorial, so they read a hundred and twenty every time, whichever candidate you are using. It is the fifth line that moves. Under nought it collapses to zero; under two it jumps to two hundred and forty; under one it stays at a hundred and twenty. Count the different values in each column and you get two, one, two.

5:37Only one candidate leaves the column standing at a single value, which is the very least you can ask of a rewriting that was supposed to change nothing. So nought factorial being one is not a convention you have to swallow. It is the only value that leaves the relation alone. The same relation lets you pull factors out one at a time, and it is worth knowing where that stops.

6:07Pull one factor off: n factorial is n times n minus one factorial. Pull two off: n times n minus one times n minus two factorial. Pull three off: n times n minus one times n minus two times n minus three factorial. Each needs room underneath, and you can find out how much by asking. Take the three-factor version at n equal to two. It asks for two times one times nought, times the factorial of minus one.

6:43There is no such thing. There is no run of whole numbers from one up to minus one, so there is nothing there to name. Ask each version for the smallest n it will say anything about, and you get one, two and three. That is where the lower bound written beside each of these comes from. It is not decoration; it is the point at which the rewriting stops naming anything.

7:12Now the reason this symbol is worth having. Seven factorial over five factorial. The obvious move is to work out both numbers and divide. Do not. Seven factorial contains five factorial whole: one, two, three, four, five, and then six and seven. Strike out the shared block on both sides. What is left above is six and seven. What is left below is nothing at all. Six times seven is forty-two, and neither factorial was ever worked out.

7:48Harder: twelve factorial, over ten factorial times two factorial. Above, the whole numbers up to twelve. Below, the whole numbers up to ten, and then one and two again. Strike out one for one. Left standing above: eleven and twelve. Left standing below: one and two. A hundred and thirty-two over two. Sixty-six. Twelve factorial is a nine-digit number, and the answer is two digits long. That is the whole saving, in one line.

8:24Two more, quickly. Five factorial over two factorial times three factorial is ten, and that shape is worth remembering, because it comes back later carrying a meaning. Nine factorial over four factorial is fifteen thousand one hundred and twenty, from five factors and no evaluation. One warning. The mark binds to the number directly in front of it and to nothing else. Seven factorial over five factorial is not the factorial of seven over five. Cancel the factors; do not tidy up the number inside the symbol.

9:07Back to nought factorial, because there is a second argument for it, and it shares no step with the first. Take four distinct objects and put all four of them in a row. Build every arrangement by hand and count them: twenty-four. Now the formula. Filling r places from n objects is n factorial over n minus r factorial. Set r equal to n, because we used everything. The bottom becomes nought factorial.

9:40If nought factorial were nought, the formula divides by nothing and hands back no answer at all. If it were two, the formula gives twelve. Half of what we counted, and we counted by hand. If it is one, the formula gives twenty-four, which is what is actually there. So two arguments, one from the relation and one from a count made by hand, corner the same value. That is what makes it forced rather than agreed.

10:12One belief worth killing before it costs you anything: the mark does not distribute across a plus sign. Three factorial is six. Four factorial is twenty-four. Add them and you have thirty. Seven factorial is five thousand and forty. Thirty against five thousand and forty: the right-hand side is a hundred and sixty-eight times the size, which is not a near miss you could put down to carelessness. Nor does it carry across a minus. Seven factorial less five factorial is four thousand nine hundred and twenty.

10:52It is not two, and it is not two factorial. Subtracting the numbers does not subtract the marks. Here is a shape that turns up constantly. One over eight factorial, plus one over nine factorial, equals x over ten factorial. Solve for x. The move is to multiply every term by the largest factorial in sight, which is ten factorial. The right-hand side becomes x, which is the whole point of choosing it.

11:28On the left, the first term becomes ten factorial over eight factorial. Strike out the shared block: nine and ten survive, so that is ninety. The second becomes ten factorial over nine factorial. Only ten survives, so that is ten. Ninety plus ten. x is a hundred. Try the same shape one size down: one over six factorial plus one over seven factorial equals x over eight factorial. Eight factorial over six factorial leaves seven and eight, so fifty-six. Eight factorial over seven factorial leaves eight.

12:12Fifty-six plus eight. x is sixty-four. A hundred, and sixty-four. A hundred is ten times ten, and sixty-four is eight times eight. And ten and eight are exactly the numbers standing under the x in each problem. So do it once in n and be finished with the whole family. One over n minus two factorial, plus one over n minus one factorial, equals x over n factorial. Multiply through by n factorial. The first quotient leaves two factors: n minus one, and n.

12:52The second leaves one factor: n. So x is n times n minus one, plus n. The minus n and the plus n cancel, and x is n times n. Run that from three up to twelve and the answers are nine, sixteen, twenty-five, thirty-six, forty-nine, sixty-four, eighty-one, a hundred, a hundred and twenty-one, a hundred and forty-four. Every one of them satisfies the equation it came from. Two problems that look like separate drill are one problem, and it has an answer you can read off without solving anything.

13:33So, the habit this all adds up to. Small factorials: work them out, and know four factorial is twenty-four on sight. Large ones sitting in a quotient: leave them alone and strike out what the two sides share. A factorial you evaluated and then had to cancel was wasted effort. And nought factorial is one because the relation and the count both refuse to work with anything else, not because someone decided it.

14:05The mark itself never was a new operation. It is a short name for the descending product the counting had already produced.