A five letter word is to be formed such that the letters appearing in the odd positions are taken from the unrepeated letters of the word MATHEMATICS whereas the letters which occupy even places are taken from amongst the repeated letters.___
A five letter word is to be formed such that the letters appearing in the odd positions are taken from the unrepeated letters of the word MATHEMATICS whereas the letters which occupy even places are taken from amongst the repeated letters.
There are 3 odd places namely 1st, 3rd and 5th which are to be filled by unrepeated 5 letters H, E, I, C, S. This can be done in 5P3 = 5.4.3 = 60 ways. We have two even places namely 2nd and 4th 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 ∴3p2=3.2=6
(ii) Both alike 3C1.2!2!=3
Thus even places can be filled in 6 + 3 = 9 ways. Hence by fundamental theorem, the required number of words is 60×9=540.
There are 3 odd places namely 1st, 3rd and 5th which are to be filled by unrepeated 5 letters H, E, I, C, S. This can be done in 5P3 = 5.4.3 = 60 ways. We have two even places namely 2nd and 4th 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 ∴3p2=3.2=6
(ii) Both alike 3C1.2!2!=3
Thus even places can be filled in 6 + 3 = 9 ways. Hence by fundamental theorem, the required number of words is 60×9=540.
