Pascal's Triangle

Transcript

0:00Introduction

Hello everyone! Okay, a small puzzle first. Eleven squared is one hundred and twenty one: one, two, one. Eleven cubed is one thousand three hundred and thirty one: one, three, three, one. Eleven to the power four: one, four, six, four, one. Rows of a famous triangle! So eleven to the power five just copies the next row, right? Hmm, not so fast. Today we meet Pascal's triangle, and its patterns come up often in JEE, so let's begin.

0:37Building the triangle

So, here is the triangle. Row n holds the binomial coefficients of x plus a, in brackets, to the power n, counting from row zero, as the labels on the left show. Row two, for example, is one, two, one. The orange band and the blue boxes come later. Okay? Now look at the two edges. Every row starts with one and ends with one, all the way down. Right? Now the inside entries. In row six, find fifteen and six, in the blue boxes. Which number sits just below them in row seven, and how is it made from them? Yes! Twenty one: fifteen plus six! The blue arrows show it: every inside entry is the sum of the two entries just above it. So you can build any row from the row above it. Clear? In symbols, this is Pascal's rule: n C r minus one, with r minus one at the bottom, plus n C r, equals n plus one C r. Remember, n C r has a small n at the top left and r at the bottom right. In the figure: six C four plus six C five equals seven C five. With me?

1:45Patterns in every row

Now the patterns. Row n lists n C zero, n C one, and so on up to n C n; the three dots mean and so on. Counted from r equals zero, that makes n plus one entries. And every row is symmetric: row six reads the same from both ends. Got it? Quick question. Add up all the entries of row four. What do you get? Correct, sixteen! And sixteen is two to the power four; the small number up top is the power. Row three adds to eight, row two to four, row one to two. So row n adds up to two to the power n, as the top line of the figure says. Right? Next, put plus and minus signs in turn. Row three: one minus three plus three minus one. Zero! This alternating sum is zero for every row after row zero. Okay? And now our puzzle. Eleven to the power n is read straight off row n, but see the condition: n less than or equal to four. Only up to row four! Why? Example one will tell us. Clear?

2:54The hockey stick

Now my favourite pattern. Look at the orange band. It runs down a diagonal: one, two, three, four, five, from row one to row five. Add them up. Where does that total sit in the triangle? Fifteen, yes! And there it is, circled in row six, just below the five and one step to the right. That is six C two. The band bends there like the blade of a hockey stick, hence the name: the hockey stick. With me? Now in general. Start at r C r on the edge, and go down the same diagonal: r plus one C r, and so on, to n C r. Where does the sum land? Very good if you said it: one row down, one step along: n plus one C r plus one. Both the top and the bottom number go up by one. Many students write n plus one C r, so watch that plus one at the bottom. Got it? And the same idea works along the other diagonal: n C zero, plus n plus one C one, and so on up to n plus m C m, equals n plus m plus one C m. One row below the last term, with the same bottom number m. Okay?

4:06Example 1: why 11^4 = 14641 but 11^5 is not 15101051

Now let's settle our puzzle. Explain why eleven to the power four is fourteen thousand six hundred and forty one, but eleven to the power five is not one, five, one, zero, one, zero, five, one. Hint: eleven is ten plus one. Pause the video and try it. Done? Okay, let's check. Eleven to the power n is ten plus one, to the power n, with n a positive integer. By the binomial theorem: n C zero, plus n C one times ten, plus n C two times ten squared, and so on. The dot means times, okay? So n C zero is the units digit, n C one the tens digit, n C two the hundreds digit. The digits are row n, read from either end, since the row is symmetric. But only while every entry is below ten. Right? Row four is one, four, six, four, one: all below ten. So eleven to the power four is fourteen thousand six hundred and forty one, straight from the row. With me? Now row five: one, five, ten, ten, five, one. Can ten sit in one place as a single digit? No, no! Ten has two digits, so writing row five side by side is wrong. We must carry, just like in addition. Clear? So write the place values in full: one times ten to the power five, plus five times ten to the power four, plus ten times ten cubed, plus ten times ten squared, plus five times ten, plus one. One hundred and sixty one thousand and fifty one! Puzzle solved. Perfect!

5:52Example 2: hockey stick proof and 3C3 + 4C3 + ... + 10C3

Now a harder one. Prove the hockey stick identity: the sum of k C r, for k from r to n, equals n plus one C r plus one. The Greek letter sigma means add up, from k equals r to n. Then find three C three, plus four C three, and so on up to ten C three. Pause the video and try it. Done? Okay, let's check. The trick is the first term: r C r is one, and r plus one C r plus one is also one. So we swap one for the other. Right? Why does that help? The next term of our sum is r plus one C r, and r plus one C r plus one sits right beside it, in the same row. So what does Pascal's rule give? Yes, exactly! The entry just below them: r plus two C r plus one. Two terms became one. With me? Now add the next term, r plus two C r: neighbours again, so we get r plus three C r plus one. Keep going, and the last step gives n plus one C r plus one. So the hockey stick is just Pascal's rule, again and again. Clear? Now the sum: three C three, four C three, up to ten C three. So r is three and n is ten. Then the answer is eleven C three, right? Oh no, careful! Both numbers go up by one. n plus one is eleven, and r plus one is four. So the sum is eleven C four. Okay? Now compute it: eleven times ten times nine times eight, over four times three times two times one. Seven thousand nine hundred and twenty, over twenty four. Three hundred and thirty! Eight terms, added in one step. Very good!

7:44Quick revision: Pascal's rule, row sums, hockey stick

So, quick revision. Each entry is the sum of the two just above it: Pascal's rule. Row n is symmetric, adds up to two to the power n, and has alternating sum zero. And the hockey stick: r C r, plus r plus one C r, and so on up to n C r, equals n plus one C r plus one. Now try the practice problems on this page. You know every pattern you need for them. All the best!