Factorials and Permutations

Transcript

0:00Introduction

Hello everyone! Okay, a small puzzle before we start. Five friends, three numbered chairs, and they want to know in how many ways they can sit. Most students say five times three, fifteen. Some say five cubed, one hundred and twenty five. Both are wrong! And here is something stranger: the number of ways to arrange nothing at all is one, not zero. By the end of this video, both will feel obvious. Permutations and factorials come up often in JEE, and the counting chapters ahead build on them. So, let's begin.

0:34Permutation: order matters

So, first the meaning. Permutation is just a fancy word for an arrangement of objects in a definite order. Swap two people between two chairs, and you get a new arrangement. Okay? Now back to our five friends. Fill chair one first. How many choices do we have? And once someone sits there, how many are left for chair two? Chair one: five choices, anyone can sit. Then four people are left for chair two, and three for chair three, right? So, by the multiplication principle from the last concept, we multiply: five times four times three. The cross on screen means times. Sixty ways! Not fifteen, and not one hundred and twenty five. Clear? And notice the conditions. The people are different. Nobody sits on two chairs, so there is no repetition. And the chairs are numbered, so order matters. With me?

1:29The formula for nPr

Now let's make it general. We have n different objects and r places, with no repetition. The sign between zero, r and n means less than or equal to, so r runs from zero up to n, okay? Same idea as the chairs. The first place has n choices, the second n minus one, the third n minus two. So, how many choices are left for place number r? n minus r plus one. Yes, exactly! Because r minus one objects are already used up. This symbol, a small n on the top left, a capital P, and a small r at the bottom right, is read n P r. The three dots mean keep going, one less each time, for r factors in all. Right? Now a new symbol, the exclamation mark. n with an exclamation mark is read n factorial: n times n minus one times n minus two, all the way down to three, two, one. The small dots between the numbers also mean times. Got it? So here is the compact form. n P r equals n factorial over the factorial of n minus r. The bottom cancels the tail of the top, and leaves exactly our r factors. With me? Quick check with the chairs. Five P three is five factorial over two factorial: one hundred and twenty over two, which is sixty. Same answer.

2:56Special cases, and repetition allowed

Okay? Now some special cases. Arrange all n objects, so r equals n. The product runs all the way down to one, so n P n is simply n factorial. Clear? But put r equals n in the compact formula, and the bottom becomes zero factorial. So tell me, what is zero factorial? Is it zero? No, no, careful! If it were zero, we would be dividing by zero. The arrow on screen means this gives: for the formula to give n factorial, zero factorial must be one. Students fall into this trap often. Right? And this solves our second puzzle. n P zero equals one: there is exactly one way to arrange nothing, which is to do nothing. Strange, but true. Got it? One last case. What if repetition is allowed, so the same object can be used again and again? How many choices does each place have then? All n, every single time! Nothing gets used up. So n times n times n, r times, which is n to the power r. The small r up there is the power, okay? So always read the question carefully for the word repetition. With me?

4:08Topper's trick: peel factorials, never expand

Now a trick that toppers use all the time. Never multiply out a big factorial. Peel it instead. n factorial equals n times the factorial of n minus one. Peel again: n times n minus one, times the factorial of n minus two. Okay? The same peeling works for n P r. n P r equals n times n minus one P r minus one. Take out the first factor, n, and r minus one factors are left. Right? So, the factorial of n plus one, over the factorial of n minus one, is just n plus one, times n. Peel the top twice, and the rest cancels completely. No big multiplication at all. Clear?

4:50Example 1: find n if nP4 = 20 times nP2

Now, Example one. Find n, if n P four equals twenty times n P two. Hint: peel n P four. Pause the video, and try it first. Done? Okay, let's check. n P four has four factors: n, n minus one, n minus two, n minus three. The first two together are exactly n P two. So n P four is n P two, times n minus two, times n minus three. With me? Cancel n P two from both sides. It is not zero, so we can. We get n minus two, times n minus three, equals twenty. Right? Expand: n squared minus five n, minus fourteen, equals zero. The small two up there means squared. It factors as n minus seven, times n plus two, so n is seven or minus two. Now think. Do we keep both? No, no! n counts objects, so it must be a natural number, at least four. So minus two is rejected, and n equals seven. Quick check: seven minus two is five, seven minus three is four, and five times four is twenty. Correct! Got it?

6:02Example 2: 4-letter arrangements from LOGARITHM

Now a harder one, JEE style. Take the word logarithm. Its nine letters are all different, okay? How many four letter arrangements can we make? How many begin with L? And how many contain L? Pause the video, and try all three. Done? Okay, let's check. Nine different letters, four places, no repetition, and order matters. So nine P four: nine times eight times seven times six, three thousand and twenty four. Right? Begin with L: fix L in the first place. Three places remain, with eight letters. So eight P three: eight times seven times six, three hundred and thirty six. Clear? Now the third part: arrangements that contain the letter L somewhere. Many students write three hundred and thirty six again. Is that right? Oh no, careful! L need not be first. It can sit in any of the four places. So count the opposite case. Without L, eight letters fill four places: eight P four, one thousand six hundred and eighty. With me? So, the ones with L are all of them, minus those without L, okay? Three thousand and twenty four minus one thousand six hundred and eighty: one thousand three hundred and forty four. Got it? And a topper always checks. L can take any of the four places, and the other three are filled in three hundred and thirty six ways. Four times three hundred and thirty six: one thousand three hundred and forty four again. Perfect!

7:32Quick revision: nPr, 0! = 1, n^r and peeling factorials

So, quick revision. n P r equals n factorial over the factorial of n minus r, for n different objects, no repetition, and order mattering. Zero factorial is one, n P n is n factorial, and with repetition allowed, it is n to the power r. And always peel factorials. Never expand them. Now go and try the practice problems on this page. You will enjoy them. All the best!