The Greatest Binomial Coefficient

Transcript

0:00Introduction

Hello everyone! Okay, a small puzzle to start. Expand one plus x, to the power ten. The coefficients are one, ten, forty five, one hundred and twenty, two hundred and ten, two hundred and fifty two, and then back down to one. The biggest sits right in the middle. Now change one plus x to one plus two x. Is the biggest coefficient still in the middle? Keep your answer in mind; by the end of this video, you will know for sure. This comes up often in JEE, so let's begin.

0:36Rise to the middle, then fall

So, first the plain binomial coefficients: n C r, the coefficients of one plus x, to the power n, with n a positive integer. On the right, they are drawn as bars, for n equals ten and for n equals eleven, with r along the bottom. See the shape? They climb to the middle, then come down the same way. Okay? Here is why. Divide each coefficient by the one before it: n C r, with the small n at the top left of the capital C and r below, over n C, r minus one. This ratio is n minus r plus one, over r. Right? Now, a coefficient beats the one before it exactly when this ratio is greater than one. So, for which r does that happen? So the coefficients keep rising while n minus r plus one, over r, is greater than one. Multiply by r, which is positive, and rearrange: that is, while r is less than n plus one, over two. With me? And once r is greater than n plus one, over two, the ratio is below one, so the coefficients fall. Exactly the shape of the bars. Clear?

1:49Even n: one peak. Odd n: two equal peaks

Now the pictures. For n equals ten, on the left, the bars rise while r is less than eleven over two, that is, five and a half. So where is the peak? At r equals five, exactly the middle, yes! So for even n, the greatest binomial coefficient is n C, n over two, with n over two written below the capital C: one single middle coefficient. Okay? Here it is ten C five, two hundred and fifty two: the orange bar on the left. Right? Now n equals eleven, on the right. Here n plus one, over two, is exactly six. So what is the ratio at r equals six: eleven minus six plus one, over six? Six over six, exactly one, correct! So the coefficient at six equals the one at five: two equally tall neighbours. So for odd n, there are two greatest binomial coefficients: n C, n minus one over two, equals n C, n plus one over two, with these fractions, in round brackets, written below. Got it? Here, eleven C five equals eleven C six: four hundred and sixty two each, the two orange bars. Even n, one peak; odd n, two equal peaks. Clear?

3:10Greatest coefficient of (a+bx)^n

Now one step further: a plus b x, to the power n, with a and b both positive. The coefficient of x to the power r, called small c r, with r written below, is n C r, times a to the power n minus r, times b to the power r. With me? Take the same ratio, c r over c, r minus one. The n C r part gives n minus r plus one, over r, as before, and the powers of a and b leave one extra factor, b over a. The dot means times. Okay? While this ratio is at least one, each coefficient is at least as big as the one before. So where does the greatest one sit? At the last step up, yes! At the largest r for which the ratio is greater than or equal to one: that is this sign, greater than with a line below it. Right? And if the ratio is exactly one at that r, then c r equals c, r minus one: two equal greatest coefficients, like the two orange bars for n equals eleven. Clear?

4:15Trap: three different kinds of greatest

Now, a very common trap. Three phrases sound almost the same. One: the greatest binomial coefficient, which is about n C r only. Okay? Two: the greatest coefficient, which also includes a to the power n minus r, and b to the power r. Got it? Three: the greatest term, which also includes x to the power r. So read the question carefully, and see which of the three it asks for. With me?

4:44Example 1: greatest binomial coefficient, n = 12 and 13

Okay, time for an example. Find the greatest binomial coefficient in one plus x, to the power twelve, and in one plus x, to the power thirteen. Pause the video and try it. Done? Okay, let's check. Twelve is even, so there is one middle coefficient: twelve C, twelve over two, that is, twelve C six, which is nine hundred and twenty four. Right? Now thirteen. Half of thirteen is six point five, and there is no coefficient number six point five! So which one is the greatest? Thirteen is odd, so there are two of them, correct! Thirteen minus one, over two, is six, and thirteen plus one, over two, is seven. Okay? So thirteen C six equals thirteen C seven: one thousand seven hundred and sixteen, the greatest value, twice. Two equal peaks, just like n equals eleven. Very good!

5:44Example 2: greatest coefficient in (1+2x)^10

Now, the question from the start. Find the greatest coefficient in one plus two x, to the power ten. Notice: greatest coefficient, not greatest binomial coefficient. Pause the video and try it. Done? Okay, first the shortcut many students take. n is ten, which is even, so the greatest is the middle one, at r equals five, right? No, no, careful! The middle rule is only for n C r alone; here the powers of two change the picture. So use the ratio, with a equal to one, b equal to two, and n equal to ten. It is eleven minus r, over r, times two: that is, two times, eleven minus r, over r. Okay? Now you solve it. For which r is this ratio at least one? Multiply by r, which is positive: twenty two minus two r is greater than or equal to r. So r is less than or equal to twenty two over three. Right? Twenty two over three lies between seven and eight, so the largest whole number r that works is seven. Got it? So the greatest coefficient is c seven: ten C seven, times two to the power seven. One hundred and twenty times one hundred and twenty eight: fifteen thousand three hundred and sixty, the coefficient of x to the power seven. Clear? Check the neighbours: c six is thirteen thousand four hundred and forty, and c eight is eleven thousand five hundred and twenty. Both smaller. Perfect! And our puzzle from the start? The middle one, c five, is ten C five times two to the power five: only eight thousand and sixty four. So no, the biggest is not in the middle any more! It sits at x to the power seven. Puzzle solved!

7:41Quick revision: middle nCr, ratio test for (a+bx)^n

So, quick revision. Even n: one greatest binomial coefficient, n C, n over two. Odd n: two equal greatest, n C, n minus one over two, and n C, n plus one over two. For a plus b x, to the power n: the greatest coefficient is at the largest r where n minus r plus one, over r, times b over a, is at least one. And binomial coefficient, coefficient and term: three different questions. Now go and try the practice problems on this page. All the best!