  Question

A five letter word is to be formed by using the letters of word MATHEMATICS such that(i) odd places of the word are to be filled with unrepresented letters and(ii) even places are to be filled with repeated letters.Then the number of words thus formed is:

A
300  B
360  C
180  D
540  Solution

The correct option is D $$540$$We have, Word MATHEMATICS The letter appearing without repetitions are H,E,C,I and S, so they can be put on odd place. $$5$$ letters can be placed on $$3$$ places $${{=}^{5}}{{P}_{3}}\,ways$$ On even places positions can be filled by the letter form M,A and T. Now, Even places can be filled in two ways. So, choose one letter from $$3$$ given letters M,A and T and arrange them in $$2!$$ Ways i.e. $$^{3}{{C}_{2}}\times 2!\,ways.$$ So, the number of ways for even places $${{=}^{3}}{{C}_{1}}{{+}^{3}}{{C}_{2}}\times 2!=9\,\,ways.$$ Hence, the required no. of ways $${{=}^{5}}{{P}_{3}}\times 9$$ $$=540\,ways.$$ Hence, this is the answer.Mathematics

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