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Competency in Focus: Combinatorics

This problem is based on Combinatorics from American Mathematics contest (AMC 10A, 2019). It includes arrangement of  \(n\)  items out of which  \(r\) items are similar.

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Next understand the problem

A child builds towers using identically shaped cubes of different color. How many different towers with a height 8 cubes can the child build with 2 red cubes, 3 blue cubes, and 4 green cubes? (One cube will be left out.) (A) 24 (B) 288 (C) 312 (D) 1260 (E) 40320
Source of the problem
American Mathematical Contest 2019, AMC 10A  Problem 17
Key Competency
Combinatorics
Difficulty Level
4/10
Suggested Book

Start with hints 

Do you really need a hint? Try it first!
Arroanging 8 Cubes is same as arranging 9 cubes and removing 1 cube from the arrangment. It will be same as arranging 8 cubes out of 9 on top of one another.
If we want to arrange the 9 things then the total no of ways are \(9!\).
But 2 cubes are red (or similiar type), 3 Blue and 4 green so we have to divide the total arrangment by \(2!\) , \(3!\) and \(4!\).

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