Out of \(15\) boys and \(15\) girls in a class, \(5\) boys and \(5\) girls got A, \(5\) boys and \(5\) girls got B, and another \(5\) boys and \(5\) girls got C on a Math-test. (There were no Ds and Fs on this test.) Determine the smallest value of \(n\in\mathbb{N}\), so that if a team of \(n\) children is set-up, then certainly at least two children of the same gender and with the same grade are on the team.
From a group of \(6\) boys and \(8\) girls we have to select a small group of \(2\) boys and \(4\) girls. How many possibilities exist for this choice?
A license plate of a car consists of \(7\) symbols so that letters are on the first three positions and digits on remaining four positions, while any used symbol can be repeated. Letters are chosen from \(26\) symbols of the alphabet and digits are chosen from the set \(\{0; 1;\dots; 9\}\). How many such license plates can be set up?
Find the number of possible ways how to organize a group of \(4\) boys and \(6\) girls in one ordered row, if all the boys should stand on the first four positions and the girls on remining positions.
There are \(5\) different roads between cities A and B. Find the number of possible ways from the city A to the city B and back, if it is required to use one road from A to B and another different one from B to A.
There are \(20\) girls and \(10\) boys in the class. How many ways are there to designate a president and vice-president of the class if it is required that at least one position will be held by a girl.