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

How many permutations can be made by using all the letters of the word $MATHEMAGICA$?

Eight chairs are numbered $1$ to $8$. Two women and three men wish to occupy one chair each. First,the women choose chairs from amongst the chairs marked $1$ to $4$,and then the men select the chairs from amongst the remaining. The number of possible arrangements is

$6$ boys and $5$ girls sit in a line such that $(I)$ no two girls sit together $(II)$ all the girls sit together. If $p$ is the number of arrangements in case $(I)$ and $q$ is the number of arrangements in case $(II)$,then $p/q =$

How many $5$-digit numbers can be formed using the digits $1$ and $2$,such that exactly one digit is different from the others?

$A$ library has $a$ copies of one book,$b$ copies of each of two books,$c$ copies of each of three books,and single copies of $d$ books. The total number of ways in which these books can be arranged is: