Let \(n\) be any whole number. Show that the product \((n+1)(n+2)\cdots(2n)\) is divisible by \(2^n\).
There are six kids in the math circle. Each kid has their own seat, and they always sit in the same one. One day, however, the head tutor decided to rearrange the seating, and it turned out that every kid ended up in a different seat from their usual one. In how many ways can the head tutor do this?
Seven students are standing on a straight line, one after the other. Three of the students, let’s call them \(A,B,\) and \(C\) behave badly and can’t be next to each other. For example: \(\star \star AB\star \star C\) and \(\star ABC\star \star \star\) are invalid arrangements, where the star denotes any other student. However, \(A\star B\star \star \star C\) is an example of a valid arrangement. How many valid arrangements are there?