In how many different ways can the letters of the word '$AUCTION$' be arranged in such a way that the vowels always come together?

In a city,no two persons have an identical set of teeth,and there is no person without a tooth. Also,no person has more than $32$ teeth. If we disregard the shape and size of the teeth and consider only the presence or absence of a tooth at each of the $32$ positions,then the maximum population of the city is:

In how many ways can $5$ prizes be distributed among four students when every student can take one or more prizes?

There are $10$ lamps in a hall. Each one of them can be switched on independently. The number of ways in which the hall can be illuminated is

$P_1$ and $P_2$ are two distinct and intersecting planes. Three non-collinear points lie on $P_1$ and another three non-collinear points lie on $P_2$ (none being on the line of intersection of the planes). Then,the maximum number of tetrahedrons formed using these six points is:

Number of different words that can be formed from all letters of the word $APPLICATION$ such that two vowels never come together is -