The number of arrangements of the letters of the word $CALCUTTA$ is:

$A$ student is to answer $10$ out of $13$ questions in an examination such that he must choose at least $4$ from the first $5$ questions. The number of choices available to him is

In how many different ways can the letters of the word '$BANKING$' be arranged so that the vowels always come together?

The exponent of $3$ in $100!$ is

The number of words (with or without meaning) that can be formed from all the letters of the word "$LETTER$" in which vowels never come together is

There are $m$ points on a straight line $AB$ and $n$ points on another line $AC$,none of them being the point $A$. Triangles are formed from these points as vertices when $(i)$ $A$ is excluded and $(ii)$ $A$ is included. Then the ratio of the number of triangles in the two cases is