Permutation - arrangement - example
Question 1 : A family of 4 brothers and 3 sisters is to be arranged in a row for a photograph. In how many ways can they be seated if all the sisters are together?
Answer: Let B
_{1},B
_{2},B
_{3},B
_{4} denote the brothers and S
_{1},S
_{2},S
_{3} denote the sisters. Since the sisters are to be seated together for a photograph, consider all the sisters as one unit or entity. Then B
_{1},B
_{2},B
_{3},B
_{4},S can be arranged to sit in 5! ways. The sisters can be arranged among themselves in 3! ways. Since the two events are independent, the total number of arrangements = 5!.3! = 720 ways.
Question 2: In how many ways can a consonant and a vowel be chosen out of the letters in the word COURAGE?
Answer: There are three consonants (C, R, G) and four vowels (A, E, O, U) in the word COURAGE.
With the consonants, we may choose any one of the 4 vowels. It can be done in 4 ways. There are three consonants.
\The total number of ways will be 4 x 3 = 12.
Question 3: How many arrangements can be made out of the letters of the word DRAUGHT, the vowels never being separated?
Answer: There are 7 letters in the word DRAUGHT, the two vowels are A and U. Since, the vowels are not to be separated, AU can be considered as one entity. Therefore, the number of letters will be 6 instead of 7. The permutations will be P(6,6) = 6! ways.
But the two vowels A and U can be arranged in two ways, i.e. AU and UA.
\The required number of arrangements = 2!.6! = 1440 ways.
Question 4:( Out of SPM Syllibus) Find the number of arrangements that can be made out of the letters i) ASSASSINATION ii) GANESHPURI.
Answer: i) The word ASSASSINATION consists of
A's = 3, S's = 4, I's = 2,
N's = 2, T's = 1, O's = 1
The total number of letters is 13 letters.
ii) The word GANESHPURI consists of 10 distinct letters.
The number of permutations is 10!.
Question 5 : ( out of SPM Syllibus) How many different arrangements can be made out of the letters in the expression a
^{3}b
^{2}c
^{4}, when written at full length?
Answer: There are 3 + 2 + 4 = 9 letters.