How many $3$-digit numbers can be formed from the digits $1, 2, 3, 4,$ and $5$ assuming that repetition of the digits is not allowed?

If the letters of the word $MOTHER$ are arranged in all possible ways and these words are listed in dictionary order,what is the rank of the word $MOTHER$?

Find the number of arrangements of the letters of the word $INDEPENDENCE$. In how many of these arrangements do the vowels never occur together?

$A$ car will hold $2$ persons in the front seat and $1$ person in the rear seat. If among $6$ persons $2$ can drive,then the number of ways in which the car can be filled is

The number of words that can be formed out of the letters of the word $ARTICLE$ so that the vowels occupy even places is

Ten chairs are numbered as $1$ to $10$. Three women and two men wish to occupy one chair each. First,the women choose the chairs marked $1$ to $6$,then the men choose the chairs from the remaining. The number of possible ways is