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.
'COMBINE'There are total 7 words. Therefore total number of possible permutations=7!There are three vowels namely O,I and E and 4 consonants namely C,M,B, and NConsider all vowels as a single set letter. Then we are left with 5 letters in total. We will permute 5 laters and later replace the single vowel letter by 3 actual vowels written adjacent to one anotherNumber of was to arrange these 5 letters is 5! also the three vowels that will come together can be arranged in 3! waysTherefore total number of permutations of given letters such that all vowels are never separated=5!×3!=720Total number of permutations in which all vowels are not together=Total number of possible permutations-total number of permutations of given letters such that all vowels are never separated=7!-720=4320___ ___ ___ ___ ___ ___ ____ 1 2 3 4 5 6 7 The vowels need to occupy odd placesTotal number of odd places are 4We choose 3 places using C34 and then the three vowels can be arranged in those three places in 3! waysTotal number of ways of placing vowels= C34×3!Consonants can be placed in remaining 4 places in 4! waysTotal number of ways of placing consonants= 4!Total number of words with vowels at odd places=4!×C34×3!=24×4×6=576
Discussion :: Permutation and Combination - General Questions (Q.No.2)
|