Problems

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?

Downtown MathHattan has a grid pattern, with \(4\) streets going east-west and \(6\) streets south-north. You take a taxi from School (A) to cinema (point B), but you would like to stop by an ice cream shop first. In how many ways can a taxi get you there if you don’t want to take a route that is longer than necessary?

Gabby the Gnome has \(3\) cloaks of different colours: blue, green, and brown. He also has \(5\) different hats: \(3\) yellow and \(2\) red. Finally, he owns \(6\) different pairs of shoes: \(2\) yellow, and \(4\) red. Gabby is selecting an outfit: a cloak, a hat, and a pair of shoes. In how many ways can he do it if he wants the colour of his shoes to match the colour of the hat?

A goofy robot named Zippity only speaks using \(0\)s and \(1\)s. Every message Zippity sends is made of \(10\) digits. How many different \(10\)-digit messages can Zippity send if each message must include exactly one run of five zeros in a row? For example, \(0011000001\) would count as a valid message, but not \(1001010001\).