How many ways can 10 letters be arranged?
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Question 407388: how many different ways can you arrange 10 letters.? Answer by jim_thompson5910(35256) You can put this solution on YOUR website! If you need more help, email me at Also, please consider visiting my website: http://www.freewebs.com/jimthompson5910/home.html and making a donation. Thank you Jim In how many ways can we arrange the English alphabet (of 26 letters) so that exactly $10$ of them lie between $A$ and $Z$? Attempt: First we select $10$ letters to put between $A$ and $Z$ in $C(24,10)$. Now the letters that lie outside get selected automatically. We consider $[A(10 letters)Z]$ as a single unit and permute this with the rest of alphabets in $15!$ ways. Letters between $A$ and $Z$ can be permuted in $10!$ ways. Finally we can also permute $A$ and $Z$ in 2 ways. So using the rule of product, the required answer would be $C(24,10)*(15!)*(10!)*2$. Is this correct? This section covers permutations and combinations. Arranging Objects The number of ways of arranging n unlike objects in a line is n! (pronounced ‘n factorial’). n! = n × (n – 1) × (n – 2) ×…× 3 × 2 × 1 Example How many different ways can the letters P, Q, R, S be arranged? The answer is 4! = 24. This is because there are four spaces to be filled: _, _, _, _ The first space can be filled by any one of the four letters. The second space can be filled by any of the remaining 3 letters. The third space can be filled by any of the 2 remaining letters and the final space must be filled by the one remaining letter. The total number of possible arrangements is therefore 4 × 3 × 2 × 1 = 4!
n!
. Example In how many ways can the letters in the word: STATISTICS be arranged? There are 3 S’s, 2 I’s and 3 T’s in this word, therefore, the number of ways of arranging the letters are: 10!=50 400 Rings and Roundabouts
When clockwise and anti-clockwise arrangements are the same, the number of ways is ½ (n – 1)! Example Ten people go to a party. How many different ways can they be seated? Anti-clockwise and clockwise arrangements are the same. Therefore, the total number of ways is ½ (10-1)! = 181 440 Combinations The number of ways of selecting r objects from n unlike objects is: Example There are 10 balls in a bag numbered from 1 to 10. Three balls are selected at random. How many different ways are there of selecting the three balls? 10C3 =10!=10 × 9 × 8= 120 Permutations A permutation is an ordered arrangement.
nPr = n! . Example In the Match of the Day’s goal of the month competition, you had to pick the top 3 goals out of 10. Since the order is important, it is the permutation formula which we use. 10P3 =10! = 720 There are therefore 720 different ways of picking the top three goals. Probability The above facts can be used to help solve problems in probability. Example In the National Lottery, 6 numbers are chosen from 49. You win if the 6 balls you pick match the six balls selected by the machine. What is the probability of winning the National Lottery? The number of ways of choosing 6 numbers from 49 is 49C6 = 13 983 816 . Therefore the probability of winning the lottery is 1/13983816 = 0.000 000 071 5 (3sf), which is about a 1 in 14 million chance. How many ways can 10 letters?There are 3,628,800 ways to arrange those letters.
How many ways can 10 people be arranged?Since these events are interlinked, you will multiply 10•9•8•7•6•5•4•3•2•1 ie 10! Thus 10 students can be arranged in a row in 3628800 ways.
How many ways can 10 books be arranged on a shelf?So, number of ways to arrange these 10 books on a shelf such that a particular pair of books is always together = 2 × 9! We know that, the number of ways of arranging 10 books on a shelf so that a particular pair is never together = 10! - (2 × 9!) = 8 × 9!
How many distinguishable arrangements of the 10 letters of the word statistics are possible?Therefore, the letters in the word "STATISTICS" can be arranged in 50400 distinct ways.
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