In how many ways can $10$ balls be divided between two boys,such that one receives $2$ balls and the other receives $8$ balls?

In an election,there are $5$ candidates and $3$ vacancies. $A$ voter can vote for a maximum of $3$ candidates. In how many ways can a voter cast their vote?

From $5$ consonants and $5$ vowels,how many words can be formed using $3$ consonants and $2$ vowels?

In an examination of $9$ papers,a candidate is successful only if they pass in more papers than they fail. In how many ways can the candidate be unsuccessful?

$A$ committee of $5$ is to be formed out of $6$ men and $4$ ladies. The number of ways this can be done,when at most $2$ ladies are included,is

The number of ways of selecting a committee of $30$ persons from $20$ boys,$20$ girls,and $20$ teachers such that the participation of the number of boys,girls,and teachers in that committee is equal,is