How many words can be made using all the letters of FACETIOU

How many words can be made using all the letters of FACETIOUSLY if one insists all six vowels remain in alphabetical order but not necessarily contiguous. Same word as in part c, but now the requirement is that \"a\" and \" e\" must be together, \"i\" and \"o\" must be together, and \"u\" and \"y\" must be together.

Solution

Dear Student Thank you for using Chegg ! a) Given word FACETIOUSLY (11 Characters) Now since we know that order of vowers are fixed No of vowels = 5 No. Of Consonants = 6 No. of permutation can be calculated Lets selectect 5 of 11 places to placethe vowels and other 6 places can be permuted in 6! Ways Total permutation = 11C5 * 6! = 332640 Solution b) a\' &\'e\' together & \'I\' & \'o\' together & \'u\' & y\' together Let us consider \'a\' & \'e\' as one entity which can be arranged in 2! Ways Similarly for \'I\' & \'o\', \'u\' & \'y\' Total Number of characters now are = 8 (Combining characters as said above so 11-3) Total Permutations = 8!*2!^3 322560 Solution