![]() Number of permutations of ‘n’ things, taken ‘r’ at a time, when a particular thing is fixed: = n-1 P r-1 ![]() Number of permutations of ‘n’ things, taken ‘r’ at a time, when a particular thing is to be always included in each arrangement ![]() The number of circular permutations of n different objects is (n-1)!.( Basic mathematics Grade XII and A foundation of Mathematics Volume II and ) Therefore the total number of ways, in this case will be 2! X 3! = 12. After fixing the position of the women (same as ‘numbering’ the seats), the arrangement of the remaining seats is equivalent to a linear arrangement. This is so because, after the women are seated, shifting the each of the men by 2 seats, will give a different arrangement. Note that we haven’t used the formula for circular arrangements now. Now that we’ve done this, the 3 men can be seated in the remaining seats in 3! or 6 ways. If each of the women is shifted by a seat in any direction, the seating arrangement remains exactly the same. We’ll first set the 3 women, on alternate seats, which can be done in (3 – 1)! or 2 ways.(We’re ignoring the other 3 seats for now. Since we don’t want the men to be seated together, the only way to do this is to make the men and women sit alternately. In how many ways can 3 men and 3 ladies be seated at around table such that no two men are seated together? Therefore the required number of ways will be 24 – 12 = 12. Similar to (i) above, the number of cases in which C & D are seated together will be 12. The total number of ways will be (5 – 1)! or 24. (ii) The number of ways, in this case would be obtained by removing all those cases (from the total possible) in which C & D are together. ![]() Therefore, the total number of ways will be 6 x 2 = 12. So effectively we’ve to arrange 4 people in a circle, the number of ways being (4 – 1)! or 6.īut in each of these arrangements, A and B can themselves interchange places in 2 ways. (i) If we wish to set A and B together in all arrangements, we can consider these two as one unit, along with 3 others. Misclleceneous Problem of Circular and Restricted permutation:Įxample:Find the number of ways in which 5 people A,B,C,D,E can be seated at a round table, such that The number of ways, when vowels being never-together= 120-28 = 92 ways. Number of ways, when vowels come together = 4! x 2!= 28 ways If all the vowels come together, then we have: (O.E.),N,T,Sīut (O,E.) can be arranged themselves in 2! ways. (iii)Two vowels (O,E,) can be arranged in the odd places (1 st, 3 rd)OR (3 rd ,5 th) or (1st,5th)= 2! x 2! x 2! ways = 8 waysĪnd three consonants (N,T,S) can be arranged in the even place (2 nd, 4 th) = 2 ! ways Hence four-letter N.O.T.S can be arranged in 4! i.e 24 ways. (ii)When ‘E’ is fixed in the middle N.O.(E),T.S. Total number of words = 3! x 2! = 12 ways. Here (NS) are fixed, hence O, T, S can be arranged in 3! waysīut (NS or SN) can be arranged themselves is 2! ways. (i) When ‘N’ and ‘S’ occupying end-places (e) TheNumber of permutations of ‘n’ things, taken all at a time, when ‘m’ specified things always come together = n ! - Įxample: How many words can be formed with the letters of the word ‘NOTES’ when: (d) TheNumber of permutations of ‘n’ things, taken ‘r’ at a time, when ‘m’ specified things always come together = m! x ( n-m+1) ! (c) TheNumber of permutations of ‘n’ things, taken ‘r’ at a time when a particular thing is never taken: = n-1 P r. (b)Number of permutations of ‘n’ things, taken ‘r’ at a time, when a particular thing is fixed: = n-1 P r-1 (a)Number of permutations of ‘n’ things, taken ‘r’ at a time, when a particular thing is to be always included in each arrangement The 8 students can be seated in a line in 8! = 40320 waysĪ Restricted permutation is a special type of permutation in which certain types of objects or data are always included or excluded and if they can come together or always stay apart. The 8 students can be seated in a circle in (8-1)! = 7! =5070 ways In such caseĮxample:In how many ways can 8 students be seated in a circle and in a line? For example, the arrangements of people in a round table. If the arrangements of objects are taken in circular order instead of a line then it is known as a circular permutation. Circular, Restricted permutation Circular Permutations:
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