How many numbers lying between $100$ and $1000$ can be formed with the digits $0, 1, 2, 3, 4, 5$,if the repetition of the digits is not allowed?

Six objects $O_1$ to $O_6$ are arranged one on top of the other. In how many ways can these be arranged such that $O_1$ and $O_2$ are the $2$ bottom-most objects?

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?

Find the number of arrangements of the letters of the word $INDEPENDENCE$. In how many of these arrangements do all the vowels always occur together?

How many numbers greater than $1000$ but not greater than $4000$ can be formed using the digits $0, 1, 2, 3, 4$ if repetition of digits is allowed?

In a circus,there are $10$ cages for $10$ animals. $4$ cages are so small that $5$ out of the $10$ animals cannot enter them. In how many ways can the $10$ animals be placed in the $10$ cages?