In how many ways can $8$ distinct toys be distributed among $5$ children?

Find the number of $4$-digit numbers that can be formed using the digits $1, 2, 3, 4, 5$ if no digit is repeated. How many of these will be even?

If $^{56}P_{r + 6} : ^{54}P_{r + 3} = 30800 : 1$,then $r = $

In how many ways can $6$ people be arranged in a row such that $A, B, C,$ and $D$ always appear in the order $ABCD$ (not necessarily together)?

If all the letters of the word $MOST$ are permuted and the words (with or without meaning) thus obtained are arranged in the dictionary order,then the rank of the word $STOM$ when counted from the rank of the word $MOST$ is:

$A$ number lock consists of three rings. If the $1^{st}$ ring is marked with digits $0$ to $9$,the $2^{nd}$ ring is marked with prime numbers greater than $2$ but less than $30$,and the $3^{rd}$ ring is marked with all vowels,find the total number of unsuccessful attempts.