Find the number of distinct arrangements of the letters of the word $CALCUTTA$.

Is $3! + 4! = 7!$?

All the letters of the word $MOTHER$ are arranged in all possible ways and the resulting words (may or may not have meaning) are arranged as in the dictionary. The number of words that appear after the word $MOTHER$ is

Find the number of ways of arranging $6$ red balls and $6$ black balls in a row such that no two black balls are together.

The number of arrangements that can be formed from the letters $a, b, c, d, e, f$ taken $3$ at a time without repetition,such that each arrangement contains at least one vowel,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 $p/q =$