Reading the pattern out of the first few expansions

0:00Squaring a two-term bracket is something you can do without thinking. Cubing one is longer, but it is still just multiplying out and collecting. Now try the same bracket to the fifth power, or the sixth, by writing out every factor and multiplying them together. It is not hard. It is simply long, and long is where mistakes live. So the question this video answers is: what do the numbers in front of those terms actually come from?

0:32Because if you know that, you never have to remember a single one of them. Start small, in a column. The bracket to the zeroth power is one. To the first power it is itself, unchanged. Squared, the numbers in front of the three terms are one, two, one. Cubed, they are one, three, three, one. And the fourth power is obtained by taking the cube and multiplying it by the bracket one more time, which gives one, four, six, four, one.

1:06Hold on to that last sentence, because it is the whole of this video. But first look at the top line again, because it is the only one of the five that carries a condition. The zeroth power is one, provided the two terms do not cancel each other out. If the two terms cancel then the bracket is nothing, and nothing raised to the zeroth power names no value at all.

1:33Twelve different pairs of values were put into the expansions at every index from one to six, and there was not a single disagreement with the bracket's own value. Five of those twelve pairs cancel to nothing, and at every index from one upward they still agree. It is only at the zeroth power that the expansion says one while the bracket itself refuses to say anything. Now read three things off that column.

2:05First, the number of terms. Count them at each index from nought to eight: one, two, three, four, five, six, seven, eight, nine. The count of terms is always one more than the index. Second, watch the two indices as you move along one expansion. The index on the first quantity falls by exactly one at every step, and the index on the second rises by exactly one. At the fifth power that happens five times over, without exception.

2:39Third, and this is the one worth keeping, add the two indices together inside any single term. They come to the index of the whole bracket, every time. Across every term of every expansion up to index eight, there is not one term where the two indices add to anything else. That third fact is the one that lets you check an expansion someone has written down, in a couple of seconds.

3:09And the reason for it is worth having, because it is not a pattern at all. The fourth power means four brackets standing side by side. To build one term of the answer you walk along those four brackets and take exactly one of the two quantities out of each. Four brackets, four choices, four letters in the term. So however many of them were the first quantity, the rest were the second, and the two counts have to add up to four.

3:40You cannot take a letter out of a bracket twice, and you cannot skip a bracket. The two indices add to the index because that is the only thing they could possibly do. Now do something slightly rude to those five lines: rub out the letters and keep only the numbers standing in front of them. One. One, one. One, two, one. One, three, three, one. One, four, six, four, one.

4:10Five rows, staggered so each one sits centred under the one above. This array carries two names. It is widely called Pascal's triangle, and it is also known as Meru Prastara, the older arrangement described by Pingala. It is the same array either way, and the interesting question is not what to call it but where the next row comes from. Here is the rule everybody is taught. Each entry on the inside of a row is the sum of the two entries standing above it.

4:47The row for index two has one entry on the inside, index three has two, and index four has three. So across those three rows there are six places the rule has anything to say about. Now, is the rule actually right? Not is it plausible. Is it right. Every row from index nought to ten was built independently, by multiplying out, and then the rule was asked to predict each row from the one above.

5:19Eleven predictions, and not one disagreement. To make sure that comparison can fail, two rules that are not the rule were scored the same way. A rule that doubles the entry above and to the left gets two of the eleven right, and both of those are rows where every neighbouring pair happens to be equal. A rule that simply copies the entry above and to the left gets one.

5:48But knowing the rule is right is not the same as knowing why. And the why has been in front of us since the second scene: the fourth power was obtained by taking the third power and multiplying it by the bracket. So take the row for index four and do exactly that. Multiplying by the bracket means multiplying by the first quantity and by the second quantity, and adding the two results.

6:17Multiplying by the first quantity raises the first index of every term by one, and leaves the numbers alone. Multiplying by the second quantity raises the second index of every term by one, and leaves the numbers alone. So you now have two copies of the row you started with, carrying exactly the same numbers. The difference is where they sit. In the row below, the first copy lands in places nought, one, two, three and four, and the second copy lands in places one, two, three, four and five.

6:55Every entry of the second copy stands exactly one place to the right of the corresponding entry of the first. So write the row down twice, slide the lower copy one step to the right, and add the columns. One, five, ten, ten, five, one. That is the addition rule, and it is not a rule about the table at all. It is a shift and an addition, and it was there in the multiplication the whole time.

7:25Done for every index from nought to ten, the shifted sum and the row built by multiplying disagree nowhere. That picture also settles something the pattern can only assert. Why is there a one at each end of every row? Count how many of the two copies reach each place of the row below. One, two, two, two, two, one. The far left is reached by the first copy alone, because the second copy starts one place further along and never gets there.

8:00The far right, by the second copy alone. Everywhere in between, both copies arrive, and adding them is the rule. So the ones at the ends are not the rule breaking down; they are the rule applied to a neighbour that is not there. In terms rather than places: the leftmost term is the one where every single bracket handed over the first quantity, and there is exactly one way for that to happen.

8:29Checked all the way to index twelve, the outermost entry of every row is one, and every place on the inside is reached by exactly two. One more thing falls out for free. Every row reads the same forwards and backwards, and that is forced rather than lucky. Swap the roles of the two quantities and the bracket turns into itself, because addition does not care which term you write first.

9:00So the expansion has to survive being read backwards, with the two indices exchanged. Do that swap on the sixth power and gather the terms again, and you get back exactly the expansion you started with. Every index from nought to twelve was checked, and there is not one row that fails to read the same both ways. Which means you only ever have to compute half of a row.

9:29Now suppose you want the fifth power. The five rows we wrote down stop at index four, so the row you need is not there yet: you have to make it. Add the neighbours along the row for index four, which is the same move as shifting and adding. One, five, ten, ten, five, one. Six entries, one more than the index, which was the first of the three things we read off.

9:56And it has four entries on its inside, so the rule reaching this row needs four arrows rather than three. So put them back, on something awkward. Take the bracket whose first quantity is two times x and whose second is three times y, and raise the whole thing to the fifth power. The row is one, five, ten, ten, five, one, and the first index runs five down to nought while the second runs the other way.

10:28Here is the only place people come apart: the whole of two-x gets raised, not just the x. Two-x to the fifth is thirty-two x to the fifth, not two x to the fifth. Keep the brackets visibly on the page and apply the exponent to what is inside them, and the six coefficients come out thirty-two, two hundred and forty, seven hundred and twenty, one thousand and eighty, eight hundred and ten, and two hundred and forty-three.

11:00The middle one is ten times eight times nine, and the one after it is ten times four times twenty-seven. Now make the slip on purpose and see what it costs. Leave the two standing in front of the x wherever a power is written, and the six coefficients become two, thirty, one hundred and eighty, five hundred and forty, eight hundred and ten, and two hundred and forty-three. Four of the six are destroyed.

11:30The two that survive are the last two, and they survive for a reason: they are the terms where the index on the first quantity is one and where it is nought, so there is no power to get wrong. A mistake that is invisible in exactly the two places you would check is a bad mistake to have. So the array works, and you know why. Here is what it costs.

11:58Suppose you want the twelfth power. The rule builds each row out of the one above it, so you must first write every row below. That is thirteen rows and ninety-one entries, and twelve separate multiplications by the bracket, before you may read off a single coefficient. The row itself, when you finally reach it, runs one, twelve, sixty-six, two hundred and twenty, four hundred and ninety-five, seven hundred and ninety-two, nine hundred and twenty-four, and then back down the same way.

12:34The array tells you where every coefficient comes from, which is worth more than a formula. But it will not let you jump. To jump straight to the coefficient you want, you need a way of naming an entry that does not mention the entry above it. That is the next thing to build, and the array is what makes it obvious why it is needed.