I would like to check if all my answers are correct. Thanks.
Find the number of ways in which the letters of the word SECTION can be arranged if
(i) the letters are not in alphabetical order,
(ii) the consonants (S, C, T, N) and vowels (E, I, O) must alternate,
(iii) all the vowels are together,
(iv) all the vowels are separated,
(v) there must be exactly two letters between the two letters E and O
Answers:
(i) $~7! = 5040$
(ii) $~ 3! \cdot 4! = 144$
(iii) $~ \displaystyle {5 \choose 1} \cdot 4! \cdot 3! = 720$
(iv) $~ \displaystyle {5 \choose 3} \cdot 3! \cdot 4! = 1440$
(v) $~ \displaystyle {4 \choose 1} \cdot 2! \cdot 2! \cdot 3! = 96~$
$~~~($group E _ _ O, then slot into the $4$ spaces$)$
Best Answer
For the first question, as letters are not in alphabetical order, the answer should be ($7! - 1$), as one of the arrangements has all letters in alphabetical order.
Second, third and fourth are correct. For the fifth question, leaving $E$ and $O$ aside, there are $5$ letters. We choose two letters to be placed between $E$ and $O$ using ${5 \choose 2}$. We can then arrange two letters in $2$ ways and we can arrange $E$ and $O$ in $2$ ways too. Finally we permute remaining $3$ letters and block of $4$ letters with $E$ and $O$ at two ends in $4!$ ways. So the answer should be,
$ \displaystyle {5 \choose 2} \cdot 2 \cdot 2 \cdot 4! = 960$.