https://www.khanacademy.org/math/ka-math-class-11/x0419e5b3b578592a:permutations-and-combinations-ncert-new/x0419e5b3b578592a:permutations-1/v/factorial 2025-03-01T15:52:37.13987578Z Factorial We finally talk about the operator Factorial. We understand its applications and solve some problems involving its simplifications. First we notice factorials, then define it while noting its notation. We then practice identifying and evaluating a few common factorials. After going through some key observations, we practice simplifying expressions and solving equations involving factorials. We finally observe that the formula for rearrangements when all objects are distinct (and also when some objects are similar) contain our beloved factorial! Ashish Gupta video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Factorial We finally talk about the operator Factorial. We understand its applications and solve some problems involving its simplifications. First we notice factorials, then define it while noting its notation. We then practice identifying and evaluating a few common factorials. After going through some key observations, we practice simplifying expressions and solving equations involving factorials. We finally observe that the formula for rearrangements when all objects are distinct (and also when some objects are similar) contain our beloved factorial! https://cdn.kastatic.org/ka-youtube-converted/FDR3AdVop0Q.mp4/FDR3AdVop0Q.mp4 412 Solving problems involving permutations of objects https://www.khanacademy.org/math/ka-math-class-11/x0419e5b3b578592a:permutations-and-combinations-ncert-new/x0419e5b3b578592a:permutations-1/v/using-and-and-or-for-letters 2025-03-01T15:52:37.13987578Z Using AND and OR for letters We now turn our attention towards rearranging letters. We use whatever we've learnt from the world of digits to solve problems involving a brand new context. We start by rearranging the letters of the word LOVE and then keep on adding constraints - first that the words start with a vowel and then that the words start and end with vowels. We practice this skill on another word NATURE - first creating a 6 lettered word and then creating a 3 lettered word. Ashish Gupta video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Using AND and OR for letters We now turn our attention towards rearranging letters. We use whatever we've learnt from the world of digits to solve problems involving a brand new context. We start by rearranging the letters of the word LOVE and then keep on adding constraints - first that the words start with a vowel and then that the words start and end with vowels. We practice this skill on another word NATURE - first creating a 6 lettered word and then creating a 3 lettered word. https://cdn.kastatic.org/ka-youtube-converted/4hmDDblM6zM.mp4/4hmDDblM6zM.mp4 315 Solving problems involving permutations of objects https://www.khanacademy.org/math/ka-math-class-11/x0419e5b3b578592a:permutations-and-combinations-ncert-new/x0419e5b3b578592a:permutations-1/v/repeated-letters 2025-03-01T15:52:37.13987578Z Repeated letters We've learnt how to rearrange MANGO. In this video, we'll find ways to rearrange APPLE. We take the example of LEAF and TREE to understand the concept. LEAF does not have repeating letters but TREE does. We figure out a way to efficiently find the number of duplicates and remove them. We use this skill for words where there are more than 2 letters repeating: ABC and AAA, FAFF etc. Finally, we summarise and try recognising the pattern that we can use for corrections in similar scenarios. Ashish Gupta video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Repeated letters We've learnt how to rearrange MANGO. In this video, we'll find ways to rearrange APPLE. We take the example of LEAF and TREE to understand the concept. LEAF does not have repeating letters but TREE does. We figure out a way to efficiently find the number of duplicates and remove them. We use this skill for words where there are more than 2 letters repeating: ABC and AAA, FAFF etc. Finally, we summarise and try recognising the pattern that we can use for corrections in similar scenarios. https://cdn.kastatic.org/ka-youtube-converted/O6SxjnjeNRc.mp4/O6SxjnjeNRc.mp4 550 Solving problems involving permutations of objects https://www.khanacademy.org/math/ka-math-class-11/x0419e5b3b578592a:permutations-and-combinations-ncert-new/x0419e5b3b578592a:permutations-1/v/multiple-repetitions 2025-03-01T15:52:37.13987578Z Multiple repetitions So far, we've learnt how to deal with words with 2 letters repeating. In this video, we'll look at ways to arrange letters of a word with more tha 2 repetitions. We start by refreshing what we've learnt about arrangements where a particular letter repeats. Through arranging letters of the words KHANACADEMY, REPETITION, and MISSISSIPPI we learn how to deal with words where there are multiple letters repeating multiple times! Ashish Gupta video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Multiple repetitions So far, we've learnt how to deal with words with 2 letters repeating. In this video, we'll look at ways to arrange letters of a word with more tha 2 repetitions. We start by refreshing what we've learnt about arrangements where a particular letter repeats. Through arranging letters of the words KHANACADEMY, REPETITION, and MISSISSIPPI we learn how to deal with words where there are multiple letters repeating multiple times! https://cdn.kastatic.org/ka-youtube-converted/lFOGNsPknVY.mp4/lFOGNsPknVY.mp4 370 Solving problems involving permutations of objects https://www.khanacademy.org/math/ka-math-class-11/x0419e5b3b578592a:permutations-and-combinations-ncert-new/x0419e5b3b578592a:permutations-1/v/bundling 2025-03-01T15:52:37.13987578Z Bundling We cover a tricky constraint in this video. How to rearrange letters of a word when a set of letters are always together. We learn how to bundle the group of letters that are constrained to always stick together. Once we've solved for the bundled group, we then figure out a systematic way to unbundle the group. We summarise the approach and then use it in another scenario where 3 vowels are constrained to be always together. We finish it off by discussing the case when the opposite is expected - that the vowels are never together. Ashish Gupta video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Bundling We cover a tricky constraint in this video. How to rearrange letters of a word when a set of letters are always together. We learn how to bundle the group of letters that are constrained to always stick together. Once we've solved for the bundled group, we then figure out a systematic way to unbundle the group. We summarise the approach and then use it in another scenario where 3 vowels are constrained to be always together. We finish it off by discussing the case when the opposite is expected - that the vowels are never together. https://cdn.kastatic.org/ka-youtube-converted/FnNUk2oRD9g.mp4/FnNUk2oRD9g.mp4 502 Solving problems involving permutations of objects https://www.khanacademy.org/math/ka-math-class-11/x0419e5b3b578592a:permutations-and-combinations-ncert-new/x0419e5b3b578592a:permutations-1/v/putting-it-all-together 2025-03-01T15:52:37.13987578Z Putting it all together We bring in all the concepts covered in the previous videos to tackle a complex rearrangement problem. We rearrange the letters of the word INDEPENDENCE. We first find all possible arrangements. We then find words which start with P. Adding another constraint, we then find words that start with P and end with I. Finally, we also look at words which are always and never together. Ashish Gupta video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Putting it all together We bring in all the concepts covered in the previous videos to tackle a complex rearrangement problem. We rearrange the letters of the word INDEPENDENCE. We first find all possible arrangements. We then find words which start with P. Adding another constraint, we then find words that start with P and end with I. Finally, we also look at words which are always and never together. https://cdn.kastatic.org/ka-youtube-converted/3sJi9ywv4cc.mp4/3sJi9ywv4cc.mp4 409 Solving problems involving permutations of objects https://www.khanacademy.org/math/ka-math-class-11/x0419e5b3b578592a:permutations-and-combinations-ncert-new/x0419e5b3b578592a:permutations-1/v/putting-it-all-together-challenge 2025-03-01T15:52:37.13987578Z Putting it all together (Challenge) We pick things up from where we left in the previous video and tackle a really hairy constraint in this one. We find the number of ways to rearrange letters of a word when two given letters are always three letters apart. For this we break down the constraint into its components - easier problems that are independently solvable. We then combine the three to get to the final answer. Ashish Gupta video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Putting it all together (Challenge) We pick things up from where we left in the previous video and tackle a really hairy constraint in this one. We find the number of ways to rearrange letters of a word when two given letters are always three letters apart. For this we break down the constraint into its components - easier problems that are independently solvable. We then combine the three to get to the final answer. https://cdn.kastatic.org/ka-youtube-converted/11PP8vw8pIQ.mp4/11PP8vw8pIQ.mp4 540 Solving problems involving permutations of objects https://www.khanacademy.org/math/ka-math-class-11/x0419e5b3b578592a:permutations-and-combinations-ncert-new/x0419e5b3b578592a:permutations-1/v/ranking 2025-03-01T15:52:37.13987578Z Ranking If all the letters of the word MATH are rearranged and put in a dictionary, what will be the rank of the word MATH? We tackle this problem and a few similar ones in this video. We realise that there are two ways to solve these problems. Slower method involves listing down all possible arrangements and then manually hunting down the word we're looking for. This is tedious and prone to errors. A much faster method is to calculate the number of words starting with first letters. This helps us navigate the dictionary neatly. Ashish Gupta video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Ranking If all the letters of the word MATH are rearranged and put in a dictionary, what will be the rank of the word MATH? We tackle this problem and a few similar ones in this video. We realise that there are two ways to solve these problems. Slower method involves listing down all possible arrangements and then manually hunting down the word we're looking for. This is tedious and prone to errors. A much faster method is to calculate the number of words starting with first letters. This helps us navigate the dictionary neatly. https://cdn.kastatic.org/ka-youtube-converted/a3vV_Rt0J70.mp4/a3vV_Rt0J70.mp4 508 Solving problems involving permutations of objects https://www.khanacademy.org/math/ka-math-class-11/x0419e5b3b578592a:permutations-and-combinations-ncert-new/x0419e5b3b578592a:permutations-1/v/and-and-or-beyond-digits-and-letters 2025-03-01T15:52:37.13987578Z AND and OR (beyond digits and letters) This video tackles the application of AND and OR beyond digits and letters. We talk about balls, coins, flags, and people. We first find the total number of arrangements of 3 types of balls. We then nominate President and Vice-president from a group of people. We find the total number of outcomes we can get if we toss a coin a specific number of times. Finally, we deal with scenarios where we create signals using 2 or more flags. Ashish Gupta video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw AND and OR (beyond digits and letters) This video tackles the application of AND and OR beyond digits and letters. We talk about balls, coins, flags, and people. We first find the total number of arrangements of 3 types of balls. We then nominate President and Vice-president from a group of people. We find the total number of outcomes we can get if we toss a coin a specific number of times. Finally, we deal with scenarios where we create signals using 2 or more flags. https://cdn.kastatic.org/ka-youtube-converted/ZDc46vjsKj4.mp4/ZDc46vjsKj4.mp4 372 Solving problems involving permutations of objects https://www.khanacademy.org/math/ka-math-class-11/x0419e5b3b578592a:permutations-and-combinations-ncert-new/x0419e5b3b578592a:permutations-1/e/factorials 2025-05-31T02:32:31.283700489Z Factorials! Factorials! Ashish Gupta exercise https://www.khanacademy.org/math/ka-math-class-11/x0419e5b3b578592a:permutations-and-combinations-ncert-new/x0419e5b3b578592a:permutations-1/e/and-or-for-letters 2025-11-18T05:20:48.433314442Z AND/OR for letters AND/OR for letters Ashish Gupta exercise https://www.khanacademy.org/math/ka-math-class-11/x0419e5b3b578592a:permutations-and-combinations-ncert-new/x0419e5b3b578592a:permutations-1/e/and-or-letters-with-repetition 2025-05-31T02:32:31.283700489Z AND/OR letters (with repetition) AND/OR letters (with repetition) Ashish Gupta exercise https://www.khanacademy.org/math/ka-math-class-11/x0419e5b3b578592a:permutations-and-combinations-ncert-new/x0419e5b3b578592a:permutations-1/e/and-or-for-words-dealing-with-multiple-repetitions 2025-05-31T02:32:31.283700489Z AND/OR for words: Dealing with multiple repetitions AND/OR for words: Dealing with multiple repetitions Ashish Gupta exercise https://www.khanacademy.org/math/ka-math-class-11/x0419e5b3b578592a:permutations-and-combinations-ncert-new/x0419e5b3b578592a:permutations-1/e/bundling 2025-05-31T02:32:31.283700489Z Bundling Bundling Ashish Gupta exercise https://www.khanacademy.org/math/ka-math-class-11/x0419e5b3b578592a:permutations-and-combinations-ncert-new/x0419e5b3b578592a:permutations-1/e/permutations-advanced 2026-03-26T13:02:58.856258642Z Spacing Permutations (challenge) Ashish Gupta exercise https://www.khanacademy.org/math/ka-math-class-11/x0419e5b3b578592a:permutations-and-combinations-ncert-new/x0419e5b3b578592a:permutations-1/e/ranking 2025-05-31T02:32:31.283700489Z Ranking Ranking Ashish Gupta exercise https://www.khanacademy.org/math/ka-math-class-11/x0419e5b3b578592a:permutations-and-combinations-ncert-new/x0419e5b3b578592a:permutations-1/e/and-or-beyond-digits-and-letters 2025-05-31T02:32:31.283700489Z AND/OR beyond digits and letters AND/OR beyond digits and letters Ashish Gupta exercise