The number of $6-$letter words, with or without meaning, that can be formed using the letters of the word $\text{MATHS}$ such that any letter that appears in the word must appear at least twice, is $. . . ..  .. $

From a class of $25$ students, $10$ are to be chosen for an excursion party. There are $3$ students who decide that either all of them will join or none of them will join. In how many ways can the excursion party be chosen?

If $n \geq 2$ is a positive integer, then the sum of the series ${ }^{ n +1} C _{2}+2\left({ }^{2} C _{2}+{ }^{3} C _{2}+{ }^{4} C _{2}+\ldots+{ }^{ n } C _{2}\right)$ is ...... .

Consider $4$ boxes, where each box contains $3$ red balls and $2$ blue balls. Assume that all $20$ balls are distinct. In how many different ways can $10$ balls be chosen from these $4$ boxes so that from each box at least one red ball and one blue ball are chosen?

A set contains $2n + 1$ elements. The number of subsets of this set containing more than $n$ elements is equal to

$^n{C_r}\,{ \div ^n}{C_{r - 1}} = $