Words from Some of the Letters

Transcript

0:00Introduction

Hello everyone! Okay, one small puzzle before we start. Take the word EXAMINATION. Eleven letters. How many four letter words can we make from its letters? Eleven letters, four places, so eleven P four: seven thousand nine hundred and twenty, right? Hmm. The true answer is less than a third of that! So where did all those words go? Today we find out. And look, words from some of the letters come up often in JEE, okay? So, let's begin.

0:34Two quick answers, both wrong

So, what went wrong? First, always list the letters. EXAMINATION has A, I and N twice each, and E, X, M, T and O once each. So only eight different letters. Right? Now, our first answer was eleven P four: the capital P, with eleven at its top left and four at its bottom right. It counts arrangements of four out of eleven different objects. But are our eleven letters all different? No! Eleven P four, seven thousand nine hundred and twenty, treats the two A's as two different letters, and the same for the I's and the N's. So the same word gets counted again and again. Too many! Clear? So, why not use only the eight different letters? Eight P four, and we are done, right? Oh no, careful! Eight P four is one thousand six hundred and eighty, but it never uses a letter twice. And EXAMINATION has two A's, so a word with two A's is allowed. Eight P four misses all such words. Too few! So the true answer lies in between. Got it?

1:44The method: split by pattern

So here is the method. First, the rules. Order matters, because a word is an arrangement. And we may use each letter at most as often as it appears in the given word. Two A's in EXAMINATION? Then at most two A's in our word. With me? Then split the words by their pattern of repetition: all letters different, one pair, two pairs, a triple, and so on. Every word has exactly one pattern, so nothing is counted twice. Okay? For each pattern, first choose the letters, then arrange them. And at the end, add up all the patterns. Choose, then arrange, then add. That's the whole method!

2:26Arranging the chosen letters

Now, the arrange part. Once the letters are chosen, some of them may repeat. And we know this already, from the lesson on repeated letters: n factorial over p factorial times q factorial. The exclamation mark is factorial, n is the number of letters, and p and q count the copies of each repeated letter. Okay? So tell me. Four letters: one pair, and two other different letters. In how many orders can they stand in a row? Four factorial over two factorial. Yes, exactly! Twenty four over two: twelve orders. Right? And two pairs? Four factorial over two factorial times two factorial, because two factorial appears once for each pair. Twenty four over four: six orders. Clear? And for three letters, one pair and one other: three factorial over two factorial, three orders. The odd letter stands first, second or third, that's all. Got it?

3:26Example 1: 3-letter words from PROPOSITION

Now let's use the method. How many three letter words can be formed from the letters of PROPOSITION? List the letters, then split by pattern. Pause the video, and try it first. Done? Okay, let's check. List first. O three times, P and I twice each, and R, S, T and N once each. Seven different letters. Okay? Now, three letters. Which patterns are possible here? All different, or one pair and one other, or all three alike. And all three alike really is possible here, because O appears three times. Right? First, all different: seven P three. The crosses mean times: seven times six times five, two hundred and ten. Clear? Now one pair and one other. Say the pair is two O's. How many choices for the third letter? Seven, right? No, no, careful! The third letter must be different from O, so only six choices. Another O would make all three alike, and that is a separate pattern. So, three choices for the pair letter, O, P or I, times six for the other letter, times three orders. Fifty four. With me? And all three alike? Only O appears three times, so the only such word is O, O, O: just one word, okay? Total: two hundred and ten, plus fifty four, plus one. Two hundred and sixty five. That's it!

5:03Example 2: 4-letter words from EXAMINATION

Now, remember our puzzle from the start? Let's finish it. How many four letter words can be formed from the letters of EXAMINATION? Pause the video, and try it. Choose, then arrange, then add. Done? Okay, let's check. We listed the letters already: eight different letters, and A, I and N appear twice each. No letter appears three times, so no triple here. Right? First, all different: eight P four, one thousand six hundred and eighty. Yes, our too few answer! It was just one pattern. Clear? Now one pair and two others. Say the pair is two A's. How many ways to choose the two other letters? They must differ from A, and from each other. Seven letters are left, so seven C two. Correct! The capital C, with seven at its top left and two at its bottom right, counts selections: twenty one. And the pair letter itself can be A, I or N: three choices. So three times twenty one: sixty three. Okay? So, sixty three words with one pair. Done, right? No, no, wait! We have only chosen the letters. Now arrange them! One pair and two others: four factorial over two factorial, twelve orders. Sixty three times twelve: seven hundred and fifty six. Choose, then arrange. Never stop halfway! Got it? Now two pairs. Choose two pair letters out of A, I and N: three C two, three ways. Each choice arranges in four factorial over two factorial times two factorial, six orders. Three times six: eighteen. With me? Total: one thousand six hundred and eighty, plus seven hundred and fifty six, plus eighteen. Two thousand four hundred and fifty four. And there is our puzzle! Not seven thousand nine hundred and twenty, not one thousand six hundred and eighty, but right in between. Mystery solved!

7:11Quick revision: split by pattern, choose, then arrange

So, quick revision. List the letters, and how often each one appears. Split by pattern: all different, one pair, two pairs, a triple, and so on. For each pattern, choose the letters, then arrange them, and add up the patterns. Now go and try the practice problems on this page. Choose, then arrange, and you will find them easy. All the best!