If ${}^{27}P_{r+7} = 7722 \times {}^{25}P_{r+4}$,then $r = $

The total number of injections (one-one into mappings) from the set $A = \{a_{1}, a_{2}, a_{3}, a_{4}\}$ to the set $B = \{b_{1}, b_{2}, b_{3}, b_{4}, b_{5}, b_{6}, b_{7}\}$ is:

How many $5$-digit numbers can be formed using the digits $0, 1, 2, 3, 5, 7$ without repetition? If all such numbers are arranged in ascending order,what is the rank of the number $70513$?

How many words,with or without meaning,each of $2$ vowels and $3$ consonants can be formed from the letters of the word $DAUGHTER$?

How many $4$-digit numbers can be formed using the digits $1, 2, 3, 4, 5, 6, 7$ such that each digit is used only once?

If $3$ books on Physics,$2$ books on Chemistry,and $4$ books on Mathematics are to be arranged on a shelf so that all the Physics books are together and all the Mathematics books are together,then the number of such arrangements is