Question

A five letter word is to be formed such that the letters appearing in the old position are taken from the unrepeated letters of the word MATHEMATICS whereas the letters which occupy even places are taken from amongst the repeated letters.

Answer 1

There are 3 odd places namely $1^{st},3^{rd}$ and $5^{th}$ which are to be filled by unrepeated 5 letters $H,E,I,C,S$. This can be done in ${}^5{P}_3=\mathrm{5.4.3}=60$ ways. We have two even places namely $2^{nd}$ and $4^{th}$ which is to be filled by repeated letters (2M, 2A, 2T) i.e. 6 letters. These two even places can be filled by 3 different types of letters as under.
(i) All different ${\therefore }^3{P}_2=3.2=6$
(ii) Both alike ${}^3{C}_1\cdot \frac{2!}{2!}=3$
Thus even places can be filled in $6+3=9$ ways. Hence by fundamental theorem, the required number of words is $60\times 9=540$.