How many words can be formed with the letters of the word $MATHEMATICS$ by rearranging them?

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?

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?

If $n, r$ are two positive integers such that $1 \leq r < n$,then ${ }^{n} P_{r+1} + r^2 { }^{n-1} P_{r-1} + (r+1) { }^{n-1} P_{r} + r { }^{n-1} P_{r-1} =$

How many distinct words can be formed using the letters of the word $EQUATION$ such that the word starts and ends with a consonant?

The number of natural numbers less than $1000$ in which no digit is repeated is