• # question_answer The number of ways in which an arrangement of 4 letters of the word 'PROPORTION' can be made is A) 700B) 750C) 758D) 800

We have got $2{{P}^{s}},\ 2{{R}^{s}},\ 3{{O}^{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 $=15\ .\ 4\ !\ =15\times 25=360$ (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 ${{10}^{5}}-252=99748$ 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 $=360+360+18+20=758$.