How many integers between $1000$ and $10000$ have no digits other than $4, 5,$ or $6$?

For a set of $5$ true-or-false questions,no student has written all the correct answers,and no $2$ students have given the same sequence of answers. What is the maximum number of students in the class,for this to be possible?

The number of words that do not start and end with vowels,formed using all the letters of the word $UNIVERSITY$ such that all vowels are in alphabetical order,is

The number of rectangles that can be obtained by joining four of $12$ vertices of a $12$-sided regular polygon is:

For a scholarship,at most $n$ candidates out of $2n+1$ can be selected. If the number of different ways of selecting at least one candidate for the scholarship is $63$,then the maximum number of candidates that can be selected for the scholarship is -

The number of arrangements of the letters of the word $SATAYPAUL$ such that no two $A$ are together and the middle letter is a consonant,is