document.write( "Question 1205565: 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? \n" ); document.write( "

Algebra.Com's Answer #842464 by math_helper(2461) $\"\"$ $\"About$

You can put this solution on YOUR website!


\n" );
document.write( "MOTIVATION\r
\n" );
document.write( "\n" );
document.write( "There is only one repeated consonant, and that is the letter T\r
\n" );
document.write( "\n" );
document.write( "The vowels are {O,I,A,I,O}
\n" );
document.write( "The remaining consonants are {M,V,N}\r
\n" );
document.write( "\n" );
document.write( "The vowels can be arranged -- while staying together in one group -- in  5!/(2!*2!) ways or  120/(2*2) = 30 ways\r
\n" );
document.write( "\n" );
document.write( "The consonants {M,V,N} can be arranged in 3! = 6 ways
\n" );
document.write( " 
\n" );
document.write( "Finally, for EACH of these vowel AND consonant arrangements, there are 4 places to put the grouped vowels: \r
\n" );
document.write( "\n" );
document.write( "    T,{vowels},{M,V,N},T
\n" );
document.write( "    T,M,{vowels},{V,N},T
\n" );
document.write( "    T,V,{vowels},{M,N},T
\n" );
document.write( "     etc.\r
\n" );
document.write( "\n" );
document.write( "This results in  30 * 6 * 4 =  ways to arrange the letters and satisfy the problem statement. \n" );
document.write( "

\n" );