Consider all the six-digit numbers that can be formed using the digits $1, 2, 3, 4, 5$ and $6,$ each digit being used exactly once. Each of such six-digit numbers have the property that for each digit, not more than two digits, smaller than that digit, appear to the right of that digit. Find the number of such six-digit numbers having the desired property.

If the number of all possible permutations of the letters of the word $MATHEMATICS$ in which the repeated letters are not together is $90\left(X\right)$, then $X=$

In how many ways the letters of word 'MACHINE' can be arranged so that vowels will always be together?

How many numbers of four digits can be made by using the digits $3, 5, 6, 7,$ and $9$?  (Repetition of digits is not allowed)