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Answer:
6 different wallpapers
Step-by-step explanation:
Given
[tex]Colours = 27[/tex]
[tex]Textures = 14[/tex]
[tex]Total = 2268[/tex]
Required
Determine the number of different wallpapers he can choose from
From the question, he can choose 1 from 27 colors;
The number of ways is:
[tex]Colours = ^{27}C_1 = \frac{27!}{(27 - 1)!*1!} = \frac{27!}{26!*1!} = = \frac{27*26!}{26!*1} = 27[/tex]
From the question, he can choose 1 from 14 textures;
The number of ways is:
[tex]Textures = ^{14}C_1 = \frac{14!}{(14 - 1)!*1!} = \frac{14!}{13!*1!} = \frac{14*13!}{13!*1} = 14[/tex]
Assume the total number of wallpapers is x and he can only select 1.
The number of ways of selection is:
[tex]Wallpapers = ^{x}C_1 = \frac{x!}{(x - 1)!*1!} = \frac{x* (x - 1)!}{(x - 1)!*1!} = x[/tex]
So, the total selection is:
[tex]Colours * Textures * Wallpapers = Total[/tex]
[tex]27 * 14* x= 2268[/tex]
[tex]378* x= 2268[/tex]
Divide both sides by 378
[tex]\frac{378* x}{378}= \frac{2268}{378}[/tex]
[tex]x= \frac{2268}{378}[/tex]
[tex]x= 6[/tex]
Hence, the number of different wallpaper is 6