Question :
8 women and 6 men are standing in a line.
(i) How many arrangements are possible if any individual can stand in any position?
(ii) In how many arrangements will all 6 men be standing next to one another?
(iii) In how many arrangements will no two men be standing next to one another?
Solution :
(i) How many arrangements are possible if any individual can stand in any position ?
Total number of members = 14
The first person can choose any one of 14 places. So he or she has 14 options.
The 2nd person has 13 options. So each person will have 1 option less than the previous person has.
= 14 ⋅ 13 ⋅ 12 ⋅ 11 ⋅ 10 ............. 1
So the total number of arrangements = 14!
(ii) In how many arrangements will all 6 men be standing next to one another?
Solution :
Let us consider all 6 men as one unit. So, 8 women can choose their places out of 9 places (14 - 6).
These 6 men can shuffle them in 6 places.
So, the number of arrangements for this case is 9! ⋅ 6!
(iii) In how many arrangements will no two men be standing next to one another?
Solution :
First, let us arrange the seats for women then, we may allot the seats for men between them.
Women are having 8 seating options. They may shuffle them into 8!.
Next considering the number of seating arrangements for men, we have 9 seats in between them. Out of these 9 seats, they may choose any 6.
So, the total number of ways = 8! ⋅ ^{9}P_{6}
Question 2 :
Find the distinct permutations of the letters of the word MISSISSIPPI?
Solution :
Since we have repeating letters, we have to use the concept given below.
Total number of letters = 11
In the word "MISSISSIPPI", the letter "S" is appearing 4 times. "I" is appearing 4 times, "P" is appearing 2 times.
= 11!/4!4!2!
= 11 ⋅ 10 ⋅ 9 ⋅ 8 ⋅ 7 ⋅ 6 ⋅ 5 ⋅ 4!/(4! 4! 2!)
= 11 ⋅ 10 ⋅ 9 ⋅ 8 ⋅ 7 ⋅ 6 ⋅ 5 / (4 ⋅ 3 ⋅ 2 ⋅ 1)(2 ⋅ 1)
= 34650
Hence the distinct permutations of the letters of the word is 34650.
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