The total number of different combinations of one or more letters which can be made from the letters of the word ‘$MISSISSIPPI$’ is

The value of $\sum \limits_{ r =0}^{20}{ }^{50- r } C _{6}$ is equal to

Ten persons, amongst whom are $A, B$ and $C$ to speak at a function. The number of ways in which it can be done if $A$ wants to speak before $B$ and $B$ wants to speak before $C$ is

An engineer is required to visit a factory for exactly four days during the first $15$ days of every month and it is mandatory that no two visits take place on consecutive days. Then the number of all possible ways in which such visits to the factory can be made by the engineer during 1$-15$ June $2021$ is. . . . . . 

If $\alpha { = ^m}{C_2}$, then $^\alpha {C_2}$is equal to

The sum $\sum\limits_{i = 0}^m {\left( {\begin{array}{*{20}{c}}{10}\\i\end{array}} \right)} \,\left( {\begin{array}{*{20}{c}}{20}\\{m - i}\end{array}} \right)\,,$ $\left( {{\rm{where}}\,\left( {\begin{array}{*{20}{c}}p\\q\end{array}} \right)\, = 0\,{\rm{if}}\,p < q} \right)$, is maximum when m is