Dividing Different Objects into Groups

Transcript

0:00Introduction

Hello everyone! Okay, one small puzzle before we start. Twelve people have to split into three groups of four. Riya counts it like this: choose four people for the first group, then four more from the eight left, and the last four go together. She multiplies and gets thirty four thousand, six hundred and fifty. But the answer key says five thousand, seven hundred and seventy five. Exactly one sixth of her number! So who is wrong, Riya or the answer key? Hold that thought. Today we find out exactly where that six comes from. In JEE, questions like this come up often. So, let's begin.

0:42Groups of different sizes

So, let's start with the simplest case. We have n different objects, all distinct, and we divide them into three groups, of sizes a, b and c. Of course, a plus b plus c must equal n, so every object goes into some group. Okay? Now tell me. In how many ways can you pick the first group, the one of size a? n C a, yes, exactly! That is the capital C, with n at its top left and a at its bottom right. Then the second group comes from the n minus a objects that are left: n minus a, C b. And the last c objects simply go together: c C c, which is just one. The small dot between them means times. With me? Now write each of these with factorials: the exclamation mark you see on screen. The middle factorials cancel, and we get n factorial, over a factorial, b factorial, c factorial. But look at the condition in the bracket: a, b and c are all different sizes. Remember that. It is the whole story of today. Clear?

1:52Equal sizes, groups not named

Now, the interesting case. Take three m different objects, and make three groups, with m objects in each. All three sizes are the same now. Right? If we pick the groups one after another, in order, like before, we get three m in brackets, factorial, over m factorial, the whole cubed. That small three at the top right means three copies of m factorial, multiplied together. Okay? But wait. Suppose you picked group one first, then group two, then group three. Your friend picks the very same three groups, but in a different order. Is that a new way of dividing? No! Correct, it is the same division. The groups have no names, so only who sits with whom matters. And three groups can be picked in three factorial, that is six, different orders. So every division was counted six times, and we divide by three factorial. Got it? In general, if k of the groups have the same size, and the groups are not named, divide further by k factorial. Groups of different sizes can never be mixed up, so for them, there is no such division.

3:02Named groups: hand them out

Right? Now change one thing. The groups now go to named people, or rooms, or teams A, B and C, as on screen. Does it matter now which group goes to whom? Yes, of course it matters! Giving the first group to one person and the second group to another is different from swapping them. So first divide, then hand the groups out. Three groups to three people: three factorial ways. The cross sign means times. Clear? So for equal sizes, look what happens. We divided by three factorial, and now we multiply by three factorial. They cancel! Three m factorial, over m factorial cubed. That is exactly our first count, picked in order. With me? And here is the classic trap. Students see equal groups and divide by k factorial every time. No, no, careful! Divide by k factorial only when the equal-sized groups are unnamed. For named groups, like people, rooms or teams, never divide by k factorial. Okay?

4:08Pairing 2n people

One special case, which comes up often: pairing. Two n people are paired off. That means n groups, each of size two, and the pairs have no names. So we get two n in brackets, factorial, over two to the power n, times n factorial. The two to the power n, with the small n at the top right, comes from two factorial for each of the n pairs. And the n factorial, because the n pairs are all the same size and not named. Got it? Now a topper's trick, for a much nicer answer. Line everyone up in any fixed order. How many choices of partner does the first person have? Two n minus one, yes! Everyone except that person. Now take the first person who is still unpaired. Two n minus three choices are left. And so on. Right? Multiply them all: one times three times five, and so on, up to two n minus one. The three dots mean and so on. Just the product of the odd numbers! Really simple to remember. Clear?

5:09Example 1: 9 books into groups of 2, 3 and 4

Now, example one. In how many ways can nine different books be divided into groups of two, three and four? Pause the video, and try it first. Done? Okay, let's check. First, before any calculation. Do we need to divide by anything here? No! Very good. The sizes two, three and four are all different, so no two groups can be mixed up. Right? So choose two books from nine: nine C two, thirty six. Then three from the seven left: seven C three, thirty five. The last four go together: four C four, just one. With me? Thirty six times thirty five is one thousand two hundred and sixty. And that is exactly nine factorial, over two factorial, three factorial, four factorial. Perfect!

6:03Example 2: 12 people into 3 groups of 4, or teams A, B, C

Now, remember Riya's puzzle from the start? Let's finish it. In how many ways can twelve people be split into three groups of four? And in how many ways into teams A, B and C, with four in each? Pause and try. Hint: ask whether the groups have names. Done? Okay, let's check. Pick the groups one after another, like Riya did. Twelve C four is four hundred and ninety five, eight C four is seventy, and four C four is one. Multiply: thirty four thousand, six hundred and fifty. Okay? So the first answer is thirty four thousand, six hundred and fifty, isn't it? No, no, wait! That is exactly Riya's mistake. The three groups are the same size, four each, and they have no names. So each division was counted three factorial times, that is six times. Twelve factorial, over four factorial, in brackets, cubed, times three factorial. That is thirty four thousand, six hundred and fifty over six: five thousand, seven hundred and seventy five. See, mystery solved: the answer key was right, and that six was three factorial! Got it? Now the second part, with teams A, B and C, okay? The teams have names, so we hand the groups out: times three factorial. Five thousand, seven hundred and seventy five times six: thirty four thousand, six hundred and fifty. So Riya's number was right after all, but only for named teams! Clear?

7:39Quick revision: unequal, equal and named groups

So, quick revision. Groups of different sizes: n factorial over a factorial, b factorial, c factorial, and no division. Equal groups with no names: k of them means divide by k factorial. Named groups, like people, rooms or teams: multiply by the ways to hand them out. Now go and try the practice problems on this page. You will find them easy, I promise. All the best!