Find the number of possible partitions of word ‘MATHEMATICS’ such that each contains atleast one vowel.For example MA-THEMATICS ,MATHEMATICS ,MATH-EMA-TICS ,MATHEMATICS are some of the possible partitions.
I caseworked on number of parts and found the answer which is $36. $But I want to know if there is any another way to solve this problem i.e. without any casework.
Each consonant must "attach" to a nearby vowel. The initial M must attach to its adjacent A, and the concluding CS must attach to their adjacent I. The final T has $3$ choices: It can attach to either the A on its left or the I on its right or to both. Likewise, the M in the middle has $3$ choices (the E on its left or the A on its right or both). Finally, the TH can either both attach to the A on their left and sever with the E or both attach to the E on their right and sever with the A, or attach to both, or sever between them, with the T attaching to the A and the H to the E, for a total of $4$ choices. This gives $3\times3\times4=36$ partitions in all.