How many words can be formed by taking $3$ consonants and $2$ vowels out of $5$ consonants and $4$ vowels?

If $^{n}C_{9} = ^{n}C_{8}$,find $^{n}C_{17}$.

There are two urns. Urn $A$ has $3$ distinct red balls and urn $B$ has $9$ distinct blue balls. From each urn,two balls are taken out at random and then transferred to the other. The number of ways in which this can be done is

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?

If the number of $5$-element subsets of the set $A = \{a_1, a_2, \dots, a_{20}\}$ containing $20$ distinct elements is $k$ times the number of $5$-element subsets containing the element $a_4$,then what is the value of $k$?

In an election,the number of candidates is $1$ more than the number of persons to be elected. If the voters can cast their votes in $254$ ways,find the number of candidates. ($A$ voter cannot cast more votes than the number of persons to be elected.)