How many permutations of the 26 letters of the English alphabet do not contain any of the strings?
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This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here! How many permutations of the 26 letters of the English alphabet do not contain any of the strings fish, rat, or bird? Please show all the steps along with necessary explanations. There are 26! permutations of the letters of the English alphabet. Let: If the word "fish" is one symbol and there are 22 other letters,
then X = 23!. If the word "rat" is one symbol and there are 23 other letters, then Y = 24!. If the word "bird" is one symbol and there are 22 other letters, then Z = 23!. We first compute for the number of permutations that contains the word fish, rat or bird: |X union Y union Z| = |X| + |Y| + |Z| - |X int Y| - |Y int Z| - |X int Z| - |X int Y int Z| Since fish and bird both has a common letter i, then |X int Z| = 0. Also, rat and bird has a common letter r, so |Y
int Z| = 0. This also corresponds to |X int Y int Z| = 0. If fish and rat are 2 separate symbols, then there are 19 other letters for permutation, so we have |X int Y| = 21!. So we have the equation: Since we need the permutations not including the given words, we must subtract this result from the total number of permutations of the English alphabet. So we have the answer: 26! - |X union Y union Z| = 26! - (23! + 24! +
23! - 21!) This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here! © BrainMass Inc. brainmass.com September 19, 2022, 10:44 am ad1c9bdddf> ADVERTISEMENTProblem 16 Easy DifficultyAnswer$402,619,359,782,336,797,900,800,000$ permutationsVideo Transcriptwere asked How many patient? 26 English alphabet. Mhm. The strings fish, right? Or birds? Other words. The strings. F I s H r a t or B I r d Find some terms here, snake. You will be the set of all patient of the 26 letters in the alphabet will define a one will be this set of all Brings painting. Yes, A two will be the set of all strings containing the word right. And a three will be the set of all strings containing the word bird. Kurtz. Let's find the magnitude of you do is the number of permutations of 26 letters. I mean letters. So we have that the magnitude of you This is going to be 26 factorial over 26 minus 26 factorial or more simply, just factorial are leaving. This one for now. Now finds the magnitude a one. Consider fish as one letter. After all, these four letters have to be next to each other in order for the string to contain the word fish. How many other letters are there? Well, we've used four. So there are 26 minus four or 22 another letter. So we're selecting 23 letters from the 23 letters. In other words, magnitude a one. This is 23 factorial over 23 minus 23 factorial gorgeous 23 factorial and again, I'll just leave it in this form. Find a magnitude eight to we use a similar procedure again, or treat rats as one letter. Then it follows that there are 26 minus three or 23 other letters. So we need to select 23 plus one or 24 letters, firmly 24 letters. So the magnitude eight to is 24 factorial over 24 minus 24 factorial or simply four factorial. And finally find the magnitude of the three. Consider Bird is one letter, and there are the word bird. Contains four letters settling. There's going to be 22 letters remaining. It's a total of 22 plus one or 23 letters. The magnitude of a three going to be 23 factorial over 23 minus three. Bacterial. Just simply 23 factorial. Alex. Consider other sets Seem to think about what about this at a one intersect, too. These are the set of all strings which contain both the word fish and the word rats. So well, consider the two words as one letter each. Then how many letters is this? Where we have f I s h, that score and our 80? It's three born. So this has money left over 26 minus seven or 19 letters. What? And therefore we want to select 19 plus two or 21 letters, probably 21 letters. So the magnitude a one intersect too is equal to 21 factorial now Likewise, we have the magnitude of a one intersect a three Well, we know that they want to respect a three contains the strings that contain both fish and bird. But think about this. If the string contains both fish and bird, then it follows that it has to contain at least two characters. I This is impossible because this is a permutation. Of the 26 letters, strings are and therefore a one intersect. A three is the empty set and it contributes zero a 268 3. These are the strings that contain both rats and birds. But again, we see then that it must contain at least two or good Rottenberg have an art. But this is impossible because this is a permutation extraneous of the 26 letters. So a 200 a three is also and also so it contributes zero. Clearly we have a one intersect A to intersect a three, obviously the whole set. Finally, we'll use the principle of inclusion and exclusion. So we have that we're looking for is the number of letters that do not contain any of the strings Fish, rabbit, bird, finest. We're going to take the total number of these strings and subtract from the number of permutations of the 26 letters that contain at least one of strings fish, rather bird. So this is the set a one union A to union A three. And the magnitude of this set is given by using the principle of inclusion and exclusion Magnitude of a one plus magnitude of a two plus magnitude. A three minus the magnitude they want intersect too, minus the magnitude they want intersect. A three minus the magnitude. A two intersecting three was magnitude a one intercept to intersect three. So we saw that the magnitude of a one This was 23 factorial magnitude of a two. This is 24 factorial, the magnitude of a three, it says. Also 20 degree factorial minus the magnitude of a one intersect a two we saw. This was 91 factorial and the rest of the terms R zero. So we'll leave this in this storm. But we saw earlier that total number of permutations Magnitude of you is 26 factorial. So our answer is going to be 26 factorial minus 23 factorial plus 24 factorial plus 23 factorial bias. It should be I don t want that and minus 21 exports. And if you plug this into a calculator, this is a huge number. This is 402 61 right? 359 72 336797 900800 000 I believe it's 402 Septal ian. 619. 6 billion 359.2 million. 782 Porcellian, 336 Brilliant. 797 billion 900 million 800,000. And these are the number of permutations. Not containing the strings bird rats or fish. Very, very large number What is the number of permutations of the 26 letters of the alphabet?26 letters can be arranged in 67,108,863 ways without any repetition of letters.
How many combinations of 26 letters are there?Answer and Explanation:
The number of possible combinations that are possible with 26 letters, with no repetition, is 67,108,863.
How many permutations of the 26 letters are there that contain neither of the sequences math nor fun?The number of unconstrained permutations of the 26 letters is 26!.
How many permutations of the alphabet are there?Note.
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