0 votes 0 votes How many number of ways are there to arrange the letters of the word "ARRANGEMENT" such that (i)there are exactly two pairs of consecutively identical letters? (ii)there are atleast three pairs of consecutively identical letters? Others discrete-mathematics + – satendra asked Nov 12, 2018 satendra 533 views answer comment Share Follow See all 7 Comments See all 7 7 Comments reply satendra commented Nov 12, 2018 reply Follow Share @aambazinga ,i think that is not correct ...can you explain how you did it? 0 votes 0 votes Shaik Masthan commented Nov 12, 2018 reply Follow Share could you give the sample output of your questions? 0 votes 0 votes srestha commented Nov 13, 2018 reply Follow Share A-2 R-2 N-2 G-1 E-2 M-1 T-1 Now, there are four pairs of such letter, which are identical (i) take any two from $4$ pairs and rest there will be 2 pairs and 3 different letters So, arrange them inside those two pairs Answer will be $\binom{4}{2}\times 2!\times \frac{4!}{2!2!}\times 3!=216$ ______________________________________________________________ (ii) Now take $3$ pairs or $4$ pairs taking $3$ pairs we can arrange them such 6 ways $R$ _ _ _ $R$ $R$ _ _ $R$ _ $R$ _ $R$ _. _ _$R$ _ _ $R$ _ _$R$ _ $R$ _$R$ _ $R$ _ So, here arrangement will be $\binom{4}{3}\times 3!\times 6=144$ And taking $4$ pairs, it will be $4!\times 3!=144$ So ans will be $144+144=288$ Plz check it 0 votes 0 votes Deepanshu commented Nov 13, 2018 reply Follow Share srestha mam i think here Answer will be (4/2)×2!×4/!2!2!×3!=216 instead of 4! /2! 2! here it should be 5! /2! 2! 0 votes 0 votes satendra commented Nov 13, 2018 reply Follow Share @shaik Please see the below link for the answer of this question: https://math.stackexchange.com/questions/2920032/inclusion-exclusion-at-least-and-exactly-arrangements 0 votes 0 votes srestha commented Nov 13, 2018 reply Follow Share @Deepanshu why? 0 votes 0 votes Deepanshu commented Nov 13, 2018 reply Follow Share srestha mam i am supposing that there are 2 pairs of consecutively letters and 3 single means g , m ,t so 5 0 votes 0 votes Please log in or register to add a comment.