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

How many $2$ digit even numbers can be formed from the digits $1, 2, 3, 4, 5$ if the digits can be repeated?

How many numbers between $999$ and $10,000$ can be formed using the digits $0, 2, 3, 6, 7, 8$ without repetition?

Eight chairs are numbered $1$ to $8$. Two women and three men wish to occupy one chair each. First,the women choose the chairs from amongst the chairs marked $1$ to $4$,and then the men select the chairs from amongst the remaining. The number of possible arrangements is

The number of arrangements of the letters of the word $SALOON$,if the two $O$'s do not come together,is

There are $9$ balls to be placed in $9$ boxes. If $5$ of the balls cannot fit into $3$ specific small boxes,find the number of ways to arrange exactly one ball in each of the boxes.