What is the total number of permutations of the letters of the word $BANANA$?

If $^n{P_r} = 720 \times {^n{C_r}}$,then $r$ is equal to:

Evaluate $7!$

Each of the $10$ letters $A, H, I, M, O, T, U, V, W$ and $X$ appears the same when looked at in a mirror. They are called symmetric letters. Other letters in the alphabet are asymmetric letters. How many $3$-letter computer passwords can be formed (no repetition allowed) with at least one symmetric letter?

How many arrangements can be made with $5$ boys and $5$ girls in a row such that boys and girls alternate?

How many distinct words can be formed using the letters of the word $ARRANGE$ such that the two $A$'s do not occur together?