The total number of $7$ -letter words that can be formed using the letters of the word $M A T H E M A T I C S$ such that the repeating letters should be placed in odd places, whereas the non-repeating letters should be placed in even places, is

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Solution

There are $4$ odd places and $3$ even places in a $7$ -letter word.
Repeating letters $- M M, A A, T T$
Non-repeating letters $- H, E, I, C, S$
Even places can be filled in $^{5} P_{3} = 60$ ways.

For odd places, there are two cases :
$(i)$ Two letters out of four are identical and two are different.
Number of ways $=^{3} C_{1} \times \frac{4!}{2!} = 36$

$(i i)$ Two letters are identical to each other and other two are also identical to each other.
Number of ways $=^{3} C_{2} \times \frac{4!}{2! 2!} = 18$
So, total number of ways to fill even places $= 36 + 18 = 54$

Hence, total number of possible words is $54 \times 60 = 3240$

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