In how many different ways can the letters of the word 'CORPORATION' be arranged so that the vowels always come together?  Choices.
A) 50400	B) 50420	C) 50500	D) 50540	E) 50590
The word 'CORPORATION' has 11 letters. It has the vowels 'O','O','A','I','O' in it and these 5 vowels should always come together. Hence these 5 vowels can be grouped and considered as a single letter. that is, CRPRTN(OOAIO).   Hence we can assume total letters as 7. But in these 7 letters, 'R' occurs 2 times and rest of the letters are different.   Number of ways to arrange these letters   =   7   !   2   !   =   7   ×   6   ×   5   ×   4   ×   3   ×   2   ×   1   2   ×   1   =   2520   =7!2!=7×6×5×4×3×2×12×1=2520   In the 5 vowels (OOAIO), 'O' occurs 3 and rest of the vowels are different.   Number of ways to arrange these vowels among themselves   =   5   !   3   !   =   5   ×   4   ×   3   ×   2   ×   1   3   ×   2   ×   1   =   20   =5!3!=5×4×3×2×13×2×1=20   Hence, required number of ways   =   2520   ×   20   =   50400   =2520×20=50400   50400)
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In how many different ways can the letters of the word 'CORPORATION' be arranged so that the vowels always come together?