In how many ways can $11$ identical pencils be distributed among $6$ children such that each child receives at least one pencil?

If the domain and range of $f(x) = ^{9-x}C_{x-1}$ contain $m$ and $n$ elements respectively,then:

The English alphabet has $5$ vowels and $21$ consonants. How many words with two different vowels and two different consonants can be formed from the alphabet?

In an election,there are $8$ candidates,out of which $5$ are to be chosen. If a voter may vote for any number of candidates but not greater than the number to be chosen,then in how many ways can a voter vote?

$A$ student has to answer $10$ out of $13$ questions in an examination,choosing at least $5$ questions from the first $6$ questions. The number of choices available to the student is:

The number of ways of distributing $500$ dissimilar boxes equally among $50$ persons is: