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The number of arrangements of the letters of the word $ARRANGEMENT$ in which two $E$s do not occur adjacently is

How many numbers can be formed using the digits $1, 2, 3, 4, 3, 2, 1$ such that odd digits always occupy odd positions?

How many distinct $9$-digit numbers can be formed by rearranging the digits of the number $223355888$ such that all odd digits occupy even positions?

How many numbers can be formed using the digits $0, 1, 2, 3, 4, 5$ that are greater than $3000$ (repetition is not allowed)?

$n$ cadets have to stand in a row. If all possible permutations are equally likely,then the probability that two particular cadets stand side by side is

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