If ${ }^{1} P_{1}+2 \cdot{ }^{2} P_{2}+3 \cdot{ }^{3} P_{3}+\ldots+15 \cdot{ }^{15} P_{15}={ }^{q} P_{r}-s$,where $0 \leq s \leq 1$,then ${ }^{q+s} C_{r-s}$ is equal to .... .

If a number is chosen at random from the four-digit numbers formed by using the digits $0, 1, 2, 3, 4, 6$ without repetition,then the probability that it is divisible by $4$ is

The value of ${}^6P_4 + 4 \cdot {}^6P_3$ is $.......$ .

If the letters of the word $INTERMEDIATE$ are arranged,what is the probability that the two $E$'s do not come together?

The number of positive integral solutions of $xyz = 60$ is

All the letters of the word $ANIMAL$ are permuted in all possible ways and the permutations thus formed are arranged in dictionary order. If the rank of the word $ANIMAL$ is $x$,then the permutation with rank $x$,among the permutations obtained by permuting the letters of the word $PERSON$ and arranging the permutations thus formed in dictionary order is