2019 AMC 10A Problems/Problem 17

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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.)

$\textbf{(A) } 24 \qquad\textbf{(B) } 288 \qquad\textbf{(C) } 312 \qquad\textbf{(D) } 1,260 \qquad\textbf{(E) } 40,320$

Solution

Arranging eight cubes is the same as arranging the nine cubes first and then removing the last cube. Every arrangement of nine cubes corresponds to another arrangement that can be used (the first eight cubes in that order and the last cube is discarded). Thus, we get 9!. However, we overcounted because the red cubes can be permuted to have the same overall arrangement and the same with the blue and green cubes. Thus, we have to divide by the $2!$ ways to arrange the red cubes $3!$ ways to arrange the blue cubes, and $4!$ ways to arrange the green cubes. Thus we have $\frac {9!} {2! * 3! * 4!}$ = $\fbox {\textbf {(D)} 1,260}$ different arrangements of towers.

- ViolinGod

Note: This is written more compactly as \[\binom{9}{2,3,4}=\binom{9}{2}\binom{9-2}{3}\binom{9-(2+3)}{4} = \boxed{1,260}\]

Solution 2

We can do this on a case-by-case basis. We divide the problem into three cases, each representing one cube to be excluded:

1) The red cube is excluded. This gives us the problem of arranging one red cube, three blue cubes, and four green cubes. The amount of arrangements would then be $\frac{8!}{4!3!}=280$. Note that we do not need to multiply by the number of red cubes because there is no way to discriminate between the first red cube and the second.

2) The blue cube is excluded. This gives us the problem of arranging two red cubes, two blue cubes, and four green cubes. The amount of arrangements would then be $\frac{8!}{2!2!4!}=420$.

3) The green cube is excluded. This gives us the problem of arranging two red cubes, three blue cubes, and three green cubes. The amount arrangements would then be $\frac{8!}{2!2!3!}=560$.

Adding up all the possibilities from cases 1, 2, and 3 above would yield $280+420+560=\fbox {\textbf {(D)} 1,260}$ as our answer.

Quick Solution

If you're running out of time, notice that A/B/C are way too small of solutions, and E would make no sense since it would be straight up 8! without restrictions - thus the answer is D. Note, not recommended.

See Also

2019 AMC 10A (ProblemsAnswer KeyResources)
Preceded by
Problem 16
Followed by
Problem 18
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 10 Problems and Solutions

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