SOLUTION: How many ways can the letters of the word “MOTIVATION” be arranged such that the first and last letters are the same, and the vowels are together?

MOTIVATION



There is only one repeated consonant, and that is the letter T



The vowels are {O,I,A,I,O}

The remaining consonants are {M,V,N}



The vowels can be arranged -- while staying together in one group -- in  5!/(2!*2!) ways or  120/(2*2) = 30 ways



The consonants {M,V,N} can be arranged in 3! = 6 ways

 

Finally, for EACH of these vowel AND consonant arrangements, there are 4 places to put the grouped vowels: 



    T,{vowels},{M,V,N},T

    T,M,{vowels},{V,N},T

    T,V,{vowels},{M,N},T

     etc.



This results in  30 * 6 * 4 =  ways to arrange the letters and satisfy the problem statement.