Assuming each letter is used only as many times as it occurs in the word. First count the ways to place vowels and consonants, without considering identity or order within their type. In addition to the two places on each ends, reserved for two consonants and two vowels respectively, there are seven places in the middle of the arrangement, which can hold three consonants and four vowels in $\frac{7!}{3!4!}$ ways. $$\oplus\oplus\,\underbrace{\oplus\oplus\oplus\otimes\otimes\otimes\otimes}_{\frac{7!}{3!4!}\text{ permutations}}\,\otimes\otimes$$ Next count the ways to fill those places. The consonants: XMTNN, can be arranged in $\frac{5!}{2!}-3!$ distinct permutations such that the pair of N do not simultaneously occupy the first two positions. The vowels: EOAAII can be arranged in $\frac{6!}{2!2!}$ distinct permutations. If the vowels in the last two places have to be distinct this would be $\frac{6!}{2!2!}-2\times\frac{4!}{2!}$ permutations. Thus there are $\frac{7!}{4!3!}\!\!\left(\frac{5!}{2!}-3!\right)\!\!\left(\frac{6!}{2!2!}\right)$ ways to arrange the letters as specified. ($340,200$) There are $\frac{7!}{4!3!}\!\!\left(\frac{5!}{2!}-3!\right)\!\!\left(\frac{6!}{2!2!}-2\times\frac{4!}{2!}\right)$ ways if the vowels in the last two places have to be distinct. ($294,840$)
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Exercise :: Permutation and Combination - General Questions
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Exercise :: Permutation and Combination - General Questions
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Exercise :: Permutation and Combination - General Questions
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How many words can be formed with letters of EXAMINATION by taking all the letters at a time in how many of these two vowels do not come together?
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