In a football championship, there were played $153$ matches. Every team played one match with each other. The number of teams participating in the championship is

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

The number of ways in which four letters of the word $‘MATHEMATICS$’ can be arranged is given by

The number of ways, in which $5$ girls and $7$ boys can be seated at a round table so that no two girls sit together, is

If $^n{C_r} = {\,^n}{C_{r - 1}}$ and $^n{P_r}{ = ^n}{P_{r + 1}}$, then the value of $n$ is

$10$ different letters of English alphabet are given. Out of these letters, words of $5$ letters are formed. How many words are formed when at least one letter is repeated