Question: The number of ways in which an arrangement of 4 letters of the word ‘PROPORTION’ can be made is
Options:
A) 700
B) 750
C) 758
D) 800
Show Answer
Answer:
Correct Answer: C
Solution:
- We have got $ 2P^{s},\ 2R^{s},\ 3O^{s},\ 1I,\ 1T,\ 1N\ i.e. $ 6 types of letters. We have to form words of 4 letters. We consider four cases (i) All 4 different: Selection $ ^{6}C_4=15 $ Arrangement $ =15\ .\ 4\ !\ =15\times 24=360 $ (ii) Two different and two alike : $ P^{s},\ R^{s} $ and $ O^{s} $ in $ ^{3}C_1=3 $ ways. Having chosen one pair we have to choose 2 different letters out of the remaining 5 different letters in $ ^{5}C_2=10 $ ways. Hence the number of selections is $ 10\times 3=30 $ . Each of the above 30 selections has 4 letters out of which 2 are alike and they can be arranged in $ \frac{4\ !}{2\ !}=12 $ ways. Hence number of arrangements is $ 12\times 30=360 $ . (iii) 2 like of one kind and 2 of other : Out of these sets of three like letters we can choose 2 sets in $ ^{3}C_2=3 $ ways. Each such selection will consist of 4 letters out of which 2 are alike of one kind, 2 of the other. They can be arranged in $ \frac{4\ !}{2\ !\ 2\ !}=6 $ ways. Hence the number of arrangements is $ 3\times 6=18 $ . (iv) 3 alike and 1 different : There is only one set consisting of 3 like letters and it can be chosen in 1 way. The remaining one letter can be chosen out of the remaining 5 types of letters in 5 ways. Hence the number of selection $ =5\times 1 $ . Each consists of 4 letters out of which 3 are alike and each of them can be arranged in $ \frac{4\ !}{3\ !}=4 $ ways. Hence the number of arrangements is $ 5\times 4=20 $ . From (i), (ii), (iii) and (iv), we get Number of selections $ =15+30+3+5=53 $ Number of arrangements $ =360+360+18+20=758 $ .
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