A five letter word is to be formed such that the letters appearing in the odd-numbered positions are taken from the letters which appear without repetition in the word MATHEMATICS. Further the letters appearing in the even-numbered positions are taken from the letters which appear with repetition in the same word MATHEMATICS. The number of ways in which the five letters word can be formed is

720

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540

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360

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Solution

The correct option is C $540$
For odd places we have 5 choices from $H, E, C, I, S$ i.e. $5 p_{3}$ ways
and for even place we have $M, A, T$
1st we choose one letter from them and then arrange them in $21$ ways i.e. $^{3} C_{2} \times 2!$
$=^{3} C_{1} +^{3} C_{2} \times 2!$
$= 9$ ways
$∴$ Total $5 p_{3} \times 9 = 540$ ways.
Hence, the answer is $540.$

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The correct option is C 540For odd places we have 5 choices from H,E,C,I,S i.e. 5p3 waysand for even place we have M,A,T1st we choose one letter from them and then arrange them in 21 ways i.e. 3C2×2!=3C1+3C2×2!=9 ways∴ Total 5p3×9=540 ways.Hence, the answer is 540.