## What are we learning ?

**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.

## First look at the knowledge graph.

## 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

Challenges and Thrills in Pre College Mathematics Excursion Of Mathematics

### 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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