The letters of the word $MODESTY$ are written in all possible orders and these words are written out as in a dictionary. Then,the rank of the word $MODESTY$ is:

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?

Find the number of distinct arrangements of the letters of the word $CALCUTTA$.

Let $b_{1} b_{2} b_{3} b_{4}$ be a $4$-element permutation with $b_{i} \in \{1, 2, 3, \ldots, 100\}$ for $1 \leq i \leq 4$ and $b_{i} \neq b_{j}$ for $i \neq j$,such that either $b_{1}, b_{2}, b_{3}$ are consecutive integers or $b_{2}, b_{3}, b_{4}$ are consecutive integers. Find the number of such permutations.

In how many ways can $5$ prizes be distributed among $4$ boys,if each boy can receive any number of prizes?

$5$ men and $4$ women are seated in a row. If the number of arrangements in which one particular man and one particular woman are together is $\alpha$ and the number of arrangements in which those two are not together is $\beta$,then $\alpha: \beta=$