$^nC_r \div ^nC_{r-1} = $

How many $3$-digit numbers can be formed by using the digits $1, 3, 6,$ and $8$,when the digits may be repeated any number of times?

If the letters of the word $MOTHER$ are permuted and all the words so formed (with or without meaning) are listed as in a dictionary,then the position of the word $MOTHER$ is

If $a$ denotes the number of permutations of $x + 2$ things taken all at a time,$b$ the number of permutations of $x$ things taken $11$ at a time,and $c$ the number of permutations of $x - 11$ things taken all at a time such that $a = 182bc$,then the value of $x$ is

There are $15$ persons in a party and each person shakes hands with every other person exactly once. What is the total number of handshakes?

In the given figure,$6$ '$A$'s should be written in such a manner that every row contains at least one '$A$'. The number of ways this is possible is: