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

Given $5$ flags of different colours,how many different signals can be generated if each signal requires the use of $2$ flags,one below the other?

How many numbers can be formed using the digits $1, 2, 3, 4, 3, 2, 1$ such that odd digits always occupy odd positions?

Find $r$ if $^{5}P_{r} = ^{6}P_{r-1}$.

$A$ dictionary is printed consisting of $7$-lettered words only that can be made using the letters of the word $CRICKET$. If the words are arranged in alphabetical order,as in an ordinary dictionary,then the number of words before the word $CRICKET$ is:

How many numbers greater than $24000$ can be formed using the digits $1, 2, 3, 4, 5$ without repetition?