The number of words that can be formed using the letters of the word $VOWEL$ such that the vowels always remain together is

There are $10$ students in a classroom,including $3$ girls $A, B,$ and $C$. In how many ways can they be arranged in a row such that no two of the three girls are together?

The number of seven-digit odd numbers that can be formed using all the seven digits $1, 2, 2, 2, 3, 3, 5$ is $.......$

How many numbers lying between $500$ and $600$ can be formed with the help of the digits $1, 2, 3, 4, 5, 6$ when the digits are not to be repeated?

$A$ dictionary is printed consisting of $7$-lettered words only that can be made using the letters of the word $CRICKET$. If the words are arranged in alphabetical order,as in an ordinary dictionary,then the number of words before the word $CRICKET$ is:

For $1 \leq r \leq n$,the value of $\frac{1}{r+1}\left\{{ }^n P_{r+1}-{ }^{(n-1)} P_{r+1}\right\}$ is equal to