The number of ways in which a committee of $6$ members can be formed from $8 $ gentlemen and $4$ ladies so that the committee contains at least $3$ ladies is

The least value of natural number $n$ satisfying $C(n,\,5) + C(n,\,6)\,\, > C(n + 1,\,5)$ is

A father with $8$ children takes them $3$ at a time to the Zoological gardens, as often as he can without taking the same $3$ children together more than once. The number of times he will go to the garden is

In a city no two persons have identical set of teeth and there is no person without a tooth. Also no person has more than $32$ teeth. If we disregard the shape and size of tooth and consider only the positioning of the teeth, then the maximum population of the city is

Two packs of $52$ cards are shuffled together. The number of ways in which a man can be dealt $26$ cards so that he does not get two cards of the same suit and same denomination is

How many words, with or without meaning, each of $2$ vowels and $3$ consonants can be formed from the letters of the word $\mathrm{DAUGHTER}$ ?