NYJC Lecture Test Q1

A shop has \(7\) mountain bicycles of different brands, \(5\) racing bicycles of different brands and \(8\) foldable bicycles of different brands on display. A cycling club wishes to buy \(6\) of these \(20\) bicycles.

1. How many different selections can be made if there must be no more than \(3\) mountain bicycles and no more than \(2\) of each of the other types of bicycles?[3]

The cycling club decides to buy \(3\) mountain bicycles, \(1\) racing bicycle and \(2\) foldable bicycles and park them on a bicycle rack, which has a row of \(10\) bicycle lots.

2. How many different arrangements can the cycle club arrange the bicycles on the bicycle rack if all the mountain bicycles are together, both the foldable bicycles are together and the empty lots are together?[2]
3. How many different arrangements are there on the cycle rack if the foldable bicycles are at each end of the bicycles and there are no spaces between any of the bicycles?[3]