Ex. 3. In how many different ways can 3 persons be seated in a coach which has 4 seats? The first person can be seated in 4 different ways. After the first person is seated, the second can be seated in 3 different ways, and finally the last person can be seated in 2 different ways. Hence the total number of ways is 4 x 3 x 2 = 24. EXERCISE 147 1. A building has 6 entrances. In how many different ways can a person enter the building and leave by a different door? 2. Between two cities 6 ferry boats are plying. In how many different ways can a man travel from one city to the other and return by a different boat? 3. A man has 5 pairs of trousers, 7 vests, and 6 coats. In how many different costumes can he appear? 4. In how many different ways may an English, a French, and a German book be selected from 6 English, 5 French, and 3 German books? 5. In how many different ways can 2 persons be seated in a coach that has 6 seats? 6. In how many different ways can 3 children be seated in 3 seats? 7. How many different words of two letters can be formed with the letters a, b, c, d, and e if the first letter is to be a vowel ? PERMUTATIONS 445. The permutations of a certain number of things are the different orders in which some or all of the things can be arranged. Thus, the permutations of the letters a, b, c, taken two at a time, are ab, ac, ba, bc, ca, cb, and their permutations, taken three at a time, are abc, acb, bac, bca, cab, cba. Ex. 1. Write all permutations that can be formed from the numbers 1, 2, 3, 4, all taken at a time. Explanation. Write first all permutations whose first place is 1, then all whose first place is 2, etc. The four columns represent the four groups thus obtained. Each group, again, is divided according to the number in the second place; thus the first column contains first the permutations commencing with 12, then those commencing with 13, and last with 14. By continuing this mode of arranging the permutations, it is easy to obtain them all. Ex. 2. Write all permutations of letters a, b, c, and d, three taken at a time. 446. The number of permutations of n different things taken at a time is denoted by the symbol "P,. Thus the number of permutations of the preceding example is 'P. This number could be obtained without writing all permutations. There are three places to be filled in by letters. The first place can be filled by a, b, c, or d, i.e. in 4 different ways (represented in 4 different columns). After the first place is filled the second place can be filled by one of the remaining letters, i.e. in 3 different ways (producing the 3 parts of each column). Similarly, the third place can be filled in 2 different ways. Hence, according to the fundamental principle, P=4 x 3 x 2 = 24. 447. To find the number of permutations of ʼn different elements taken at a time. There are r places to be filled. The first place can be filled in any of the n ways, and after this has been filled, the second place can be filled in (n-1) ways. Hence, according to the fundamental principle, the two places together can be filled in n (n − 1) different ways; or After the first two places are filled, the third one can evidently be filled in (n-2) different ways; or By continuing this process, it can be seen that "P, is equal to a product of r factors, the first factor being n, and each following factor being less by one than the preceding one. Since the last factor must be (n − r+1), we have 448. The number of permutations of n elements, taken all at a time, is NOTE. Unless stated otherwise, the things are supposed to be different and not to be repeated in one permutation. Ex. 1. How many different permutations can be made by taking 4 of the letters of the word fraction? Ex. 2. How many numbers between 1000 and 10,000 can be formed with the figures 1, 2, 3, 4, 5, 6, 7, no figure being repeated? Since numbers between 1000 and 10,000 have four places, the required number is "P1 = 7.6.5.4 = 840. EXERCISE 148 1. Write all permutations of the numbers 1, 2, 3, taken all at a time. 2. Write all permutations of the letters a, b, c, and d, taken all at a time. 3. Write all permutations of the letters a, b, c, d, taken two at a time. 4. Write all permutations of the numbers 1, 2, 3, 4, 5, taken two at a time. 5. Find the value of 5P, P1, 10 P 6. Find the value of P2, 5P, "P 7. In how many different ways can 5 pupils be seated in 5 seats? 8. In how many different ways can 5 pupils be seated in 6 seats? 9. How many different words can be formed with the letters of the word equation? 10. How many different numbers of three different figures can be formed from the digits 2, 3, 4? 11. How many different numbers of four different figures can be formed from the digits 1, 2, 3, 4, 5, 6? 12. How many different words of three letters can be formed from the letters a, b, c, e? 13. In how many different ways can 6 persons be placed in 6 seats? 14. Six persons enter a car in which there are 10 seats. In how many different ways may they take their places? 15. How many different arrangements can be made by taking 3 letters of the word theory? 16. How many three-lettered words can be made from 10 letters, no letter being repeated in the same word? 17. In how many different 18. If "P=4"P2, find n. ways can 5 persons form a ring? 449. To find the number of permutations of things which are not all different, taking them all at a time, let us suppose the number of permutations which can be formed by taking all the letters a, a, a, b, c, was x. If we should replace the three equal a's by three different letters, as a1, a2, α, the number of permutations would obviously be greater than a, for in each of the x permutations we could arrange the a, a, a in [3 different ways without changing the position of the b and c. Thus, abaac would produce the 6 permutations, a ba,a,c, a,bajaӡс, азвајас, а baza,c, a,bazajc, agba a c. Similarly, every one of the x permutations would produce 3 arrangements, i.e. the total number of arrangements would But, on the other hand, this number represents the number of permutations of 5 elements, all being different, or 5. |