This section covers permutations and combinations. Show
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.
Continue Learning about Other Math How could you find the number of different combinations of six letters in a computer password?When trying to work out how many different combinations there are, you need to know how many options there are for each value. If the password only contains lower case letters, then we have 26 options for each value. For each letter in the password, there are 26 options, so the total number of possible options is 26x26x26x26x26x26 or 266 This equals 308,915,776 so there are 308,915,776 possible different combinations of six letters. How many different 2 number combinations are with 243?Three combinations: 23, 24 and 34 A keycode must contain 2 letters and 3 numbers. The letters may be any letter of the alphabet. The numbers should be any number from 0 to 9. How many different keycode combinations are there?676,000 but I don’t know why How many possible combinations of three letters are there?Say you have the letters A,B, and C. Here are all the possible combinations. * ABC * ACB * BAC * BCA * CAB * CBA So, 6 if you don't repeat any of the letters. If you DO repeat letters, then simply take the number of letters you have, (3 for instance), and multiply it to the power of the number of letters you have. So, for 3 letters, the formula would be 33 . Or if you had 4 letters it would be 44 and so on. If using 7 single digit numbers how many different combinations can you get?Number of 7 digit combinations out of the 10 one-digit numbers = 120. How many permutations of 6 letters are there?So there are \(6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 720\) permutations of the 6 letters.
How many combinations can 6 characters make?Continuing in this fashion you can see that there are 366 = 2 176 782 336 ways to fill in all 6 characters.
How many possible 6 letter words are there?How many 6 letter words are there? There are more than 20,000 six-letter words in the Official Scrabble Player's Dictionary, Volume 6.
How many products can be made with 6 letters?The total number of letters is 6. ⇒ 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720.
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