How many ways can a letter be arranged?
Exercise :: Permutation and Combination - General Questions11. In how many ways can a group of 5 men and 2 women be made out of a total of 7 men and 3 women? Show Answer: Option A Explanation: Required number of ways = (7C5 x 3C2) = (7C2 x 3C1) =  But, not all of the letters are unique. There are 2 f's, and there are 2 e's. Of all the 9! arrangements above, some of them are essentially the same, just with the f's (and/or e's) swapped around. For example,
would both be counted in the 9!, but they're both the same word. We need to remove all the "double-countings". This is done by dividing 9! by the number of ways to shuffle each copied letter. Since there are two F's, they can be shuffled 2! ways, and so can the two E's:
Another example: How many ways can the letters in BANANA be arranged? There are 6 letters total, with 3 A's and 2 N's. Thus, the number of unique arrangements (or "words") is: If we had 6 unique letters, such as STABLE, we'd be able to arrange the letters in#6! = 720#ways (we'd have 6 choices of what the first letter could be, 5 for the next letter, 4 for the next, etc... giving#6xx5xx4xx3xx2xx1 = 6!# In our case, we have 2 sets of letters where there are more than 1 - we have two S's and two T's. And so we have to divide out the number of ways each of them can order#(S_1TATES_2#is the same as#(S_2TATES_1)#and so we'll have: 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 a letter can be arranged?=360 the number of ways.
How many ways can the letters ABCD be arranged?Total possible arrangement of letters a b c d is 24. together. as one entity so we have total 3 letters. 3 letters can be arranged in 3!
How many ways can 4 letters be arranged?= 4 * 3 * 2 *1 = 24 ways to arrange four letters.
How many ways are there to rearrange the letters in function?(2*3*4*5*6*7*8/2) (3*4*5*6*7*8) = 20160. There is 20160 ways we can rearrange lettes in Word “Function”.
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