Tower from cubes
Task number: 4493
How many ways are there to build a tower with 5 green, 4 red and 3 blue cubes? Cubes of the same color are indistinguishable from each other.
Solution
Five of the twelve positions in the tower have green cubes, that is \(\binom{12}{5}=792\) possibilities. If we remove them, we get a tower of seven pieces and now have four positions for red cubes, or \(\binom{7}{4}=35\) possibilities.
In fact, each position of the tower corresponds to a twelve letter word, with five letters G representing the position of the green cubes, four letters R and three letters B. There are \(\binom{12}{5{,}4,3}=\frac{12!}{5!4!3!}\) of such words.
Answer
The tower can be built in \(27 720\) ways.
