Factorials and Binomial Coefficients

Transcript

0:00Introduction

Hello everyone! Okay, a small puzzle before we start. Fifteen C three r equals fifteen C r plus three. Find r. Many students just match the bottoms: three r equals r plus three, so r is three over two. One and a half objects chosen? Hmm, something is not right here, is it? By the end of this video, you will find the real answer in two lines. These symbols, n C r, are the binomial coefficients, and the whole binomial theorem is built on them. They come up often in JEE, so let's begin.

0:37Factorials: the building block

So, first, what is a binomial? Really simple: a sum of two terms, like x plus a, two x minus three y, or one plus x. Okay? The binomial theorem writes x plus a, whole to the power n, as a sum of simple terms. And the numbers in front of those terms are the binomial coefficients. So we start with them, right? To build them, we need the factorial. n factorial, written n with an exclamation mark, is one times two times three, and so on, up to n. The raised dots mean times; the three dots mean and so on. And remember: zero factorial is one, not zero. Clear?

1:18The binomial coefficient nCr

Now the star of today's class: n C r. See the symbol: a small n on top at the left, a capital C, and a small r at the bottom on the right. It is also written as n over r, in tall round brackets. Its value: n factorial, over r factorial times the factorial of n minus r. n and r are integers, and r goes from zero to n: those are less than or equal to signs. With me? Now, n factorial over the factorial of n minus r: the whole factorial of n minus r cancels from the top. So tell me, how many factors stay on top? Exactly r factors! n, times n minus one, times n minus two, all the way down to n minus r plus one, over r factorial. That is the fast way in the exam. Okay? And what does n C r mean? It is the number of ways of choosing r objects out of n different objects. Keep this meaning in mind; we will use it soon. Right? And one small convention: if r is less than zero, or greater than n, we take n C r as zero. You cannot choose more objects than you have. Makes sense?

2:33Quick values and symmetry

Now a few values to know by heart. n C zero and n C n are both one. n C one and n C n minus one are both n. Okay? And n C two is n times n minus one, over two. You will use this one a lot; it saves time. Got it? Did you notice? n C zero equals n C n. That is symmetry: n C r equals n C n minus r. Why? Put n minus r in place of r, and the bottom, r factorial times the factorial of n minus r, just swaps its two factors. Same value! Clear? Now, a question that comes up often. If n C x equals n C y, must x equal y? No, no, not necessarily! Either x equals y, or x plus y equals n. The double arrow means: this gives. Forgetting the second case is exactly the trap in our puzzle. Right?

3:31Pascal's rule and absorption

Next, Pascal's rule. n C r plus n C r minus one equals n plus one C r. Same top, bottoms side by side, and they add up to one coefficient with top n plus one. With me? But why is this true? Use the meaning of n C r: choosing r people out of n plus one people. Can you split those choices into two groups? Yes, two groups! Fix one person, call that person P, okay? Committees without P: choose all r from the other n, so n C r. Committees with P: P already has a seat, so choose r minus one from n, that is n C r minus one. Together, every committee: n plus one C r. Beautiful, isn't it? Now absorption: r times n C r equals n times n minus one C r minus one. The r in front becomes an n, and the top and the bottom both drop by one. Clear?

4:31Topper's trick: the ratio of neighbours

Now, the topper's trick. Take two neighbours: n C r over n C r minus one. It equals n minus r plus one, over r. This is the most used line in this whole unit. Okay? How to remember it? Top minus bottom plus one, over bottom. The top is n, the bottom is r. So, n minus r plus one, over r. Really simple! Right? And where do we use it? Consecutive coefficients, the greatest coefficient, the greatest term: all solved with this one line. You will see it in Example two. With me?

5:09Example 1: solve 15C(3r) = 15C(r+3)

Now, remember our puzzle from the start? Let's finish it. Solve fifteen C three r equals fifteen C r plus three. Remember, there are two cases. Pause the video and try it. Done? Okay, let's check. Case one: the bottoms are equal. Three r equals r plus three, which gives r equals three over two. So the answer is three over two, right? Oh no, careful! n C r is defined only for integers. Three over two is not an integer, so this case is rejected. That was the one and a half objects in our puzzle! Clear? So, case two: the bottoms add up to the top. Three r plus r plus three equals fifteen. Four r equals twelve, so r equals three. Okay? Let's check. Three r is nine, r plus three is six, and nine plus six is fifteen. So fifteen C nine equals fifteen C six, by symmetry. Perfect! Puzzle solved, and r is three.

6:12Example 2: find n and r from 36, 84 and 126

Now a harder one. Three neighbours: n C r minus one is thirty six, n C r is eighty four, and n C r plus one is one hundred and twenty six. Find n and r. Pause the video and try it. Alright. Three neighbouring coefficients, and two unknowns. Which line from today will crack this? Yes, exactly! The ratio of neighbours. n C r over n C r minus one is n minus r plus one, over r. That equals eighty four over thirty six, so seven over three. Cross multiply: three n plus three equals ten r. Okay? Now the next pair: n C r plus one, over n C r. Careful, the bottom is now r plus one. So what is the ratio? Top minus bottom plus one: n minus r minus one plus one, that is n minus r, over r plus one. It equals one hundred and twenty six over eighty four, that is three over two. So two n equals five r plus three. Right? Now solve. From the second, n equals five r plus three, over two. Put this into the first: fifteen r plus nine, over two, plus three equals ten r. With me? Multiply by two: fifteen r plus fifteen equals twenty r. So r equals three, and putting it into five r plus three, over two, n is nine. Correct! Always check. Nine C two is thirty six, nine C three is eighty four, and nine C four is one hundred and twenty six. All three match. Very good!

7:55Quick revision: nCr formula, nCx = nCy, neighbour ratio

So, quick revision. n C r is n factorial, over r factorial times the factorial of n minus r, and zero factorial is one. If n C x equals n C y, then x equals y, or x plus y equals n. Never forget the second case. And neighbours: top minus bottom plus one, over bottom. Now try the practice problems on this page. All the best!