An office party is to have as part of its decorations 10 ceiling-to-floor vertic
ID: 2902226 • Letter: A
Question
An office party is to have as part of its decorations 10 ceiling-to-floor vertical strips of colored paper bunting hanging side-by-side on a wall. Alan's boss has asked him to get the bunting (as economically as possible), but when Alan asks what colors, the boss says: I haven't decided yet, but I want there to be at least 10,000 different arrangements for me to choose from! If Alan wants to economize by having a few different colors as possible, but also wants to offer his boss the required number of choices, what is the fewest number of colors he can use, and how many strips of each color should he buy?
Explanation / Answer
You cannot do this using 2 colors since maximum arrangments are 210 = 1024
so you need atleast 3 and with three if you buy 10 sheets of each color we'll have 310 = 59,049 which are redundant so we'll not buy 10strips of each color instead
so you need to buy atleast 10strips let them be 3 red, 3 blue & 4 green
With this the number of arrangements are 10!/ 3!3!4! = 4200
so lets take an extra blue strip
so now we have 4 blue, 4 green and 3 red strips which give 3 options
option1: take 3B, 4G, 3R in to consideration we'll get 4200
option2: take 4B, 3G, 3R in to consideration we'll get another 4200
option3: take 4B, 4G, 2R in to consideration we'll get 3150 (10!/2!4!4!)
So you just need 11 strips of them we have 4Blue, 4 green and 3 red
So total no of different buntings will be more than 10000 with just 11 strips