Problems

Mary, who wrote an online computer game, plans to assign every user a unique password.
a) Currently, she wants to have passwords that are $8$ symbols long and are all made from lowercase letters $a,b,c$. How many different passwords can she generate?
b) Her friend suggested a modification: use 8-symbol words made from letters $a,b,c$, with exactly one capital letter, and also include one digit ($0-9$). How many times more passwords can she generate now?

How many numbers in the set $\{1,2,3,\dots ,1000\}$ can be expressed as a sum of $6$ consecutive integer numbers?

How many $7$-digit numbers, larger than $6$ million, are there such that the product of their digits is $42$?

On a sheet of paper a grid of $n$ horizontal and $n$ vertical straight lines is drawn. How many different closed $2n$-link broken lines can be drawn along the grid lines so that each broken line passes through all horizontal and all vertical straight lines? On the diagram below you can see an example of a closed broken line for $n=5$.

Find the largest number $A$ such that for each permutation of the set $\{1,2,3,\dots ,100\}$, the sum of some $10$ consecutive terms of that permutation is at least equal to $A$.

The ant crawls a closed path along the edges of the dodecahedron, not turning back anywhere. The path runs exactly twice on each edge. Prove that the ant crawls on some edge of the dodecahedron in the same direction both times.

Given a pile of cards, is it true that reversing the order of the pile by counting the cards out one by one leaves no card in its original position?

How many permutations of $13$ cards swap the third card and the fourth card?

Consider lifting the top $2$ cards from a pile of $6$ cards and placing them at the bottom of the pile. Show how the same process can be done by switching adjacent pairs of cards.

Lennart is a skilled magician - he can shuffle a deck of card in the most arbitrary ways and replicate the shuffle. Once he performed a terribly complicated shuffle and repeated it right away. It turned out all the cards returned to their original position. Show that this complicated shuffle consists of only switching distinct pairs of cards (or keeping some cards where they are).