Evaluate $8!$.

The sum of all $4$-digit numbers that can be formed by using the digits $2, 4, 6, 8$ (repetition of digits not allowed) is

Find the number of permutations of the letters of the word $ALLAHABAD$.

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

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

The numbers $1, 2, 3,$ and $4$ are written separately on four slips of paper. The slips are put in a box and mixed thoroughly. $A$ person draws two slips from the box,one after the other,without replacement. Describe the sample space for the experiment.