Alice the fox and Basilio the cat have grown \(20\) counterfeit bills on a money tree and now write seven-digit numbers on them. Each bill has \(7\) empty cells for numbers. Basilio calls out one digit "1" or "2" (he doesn’t know the others), and Alice writes the number into any empty cell of any bill and shows the result to Basilio. When all the cells are filled, Basilio takes as many bills with different numbers as possible (out of several with the same number, he takes only one), and the rest is taken by Alice. What is the largest number of bills Basilio can get, regardless of Alice’s actions?
There are six letters in the alphabet of the Bim-Bam tribe. A word is any sequence of six letters that has at least two identical letters. How many words are there in the language of the Bim-Bam tribe?
In how many ways can eight rooks be arranged on the chessboard in such a way that none of them can take any other. The color of the rooks does not matter, it’s everyone against everyone.
A coin is tossed six times. How many different sequences of heads and tails can you get?
Each cell of a \(3 \times 3\) square can be painted either black, or white, or grey. How many different ways are there to colour in this table?
Consider a set of numbers \(\{1,2,3,4,...n\}\) for natural \(n\). Find the number of permutations of this set, namely the number of possible sequences \((a_1,a_2,...a_n)\) where \(a_i\) are different numbers from \(1\) to \(n\).
There are two piles of rocks, \(10\) rocks in each pile. Fred and George play a game, taking the rocks away. They are allowed to take any number of rocks only from one pile per turn. The one who has nothing to take loses. If Fred starts, who has the winning strategy?
A group of \(15\) elves decided to pay a visit to their relatives in a distant village. They have a horse carriage that fits only \(5\) elves. In how many ways can they assemble the ambassador team, if at least one person in the team needs to be able to operate the carriage, and only \(5\) elves in the group can do that?
There are \(5\) pirates and they want to share \(8\) identical gold coins. In how many ways can they do it if each pirate has to get at least one coin?
We want to wrap \(12\) Christmas presents in different coloured paper. We have \(6\) different patterns of paper and we want to use each one exactly twice. In how many ways can we do this?