Answer to Question #150266 in Combinatorics | Number Theory for Asfand Khan

Question

Vasya has cubes of three colors. He builds a tower from them, placing each next cube on the previous one. It is forbidden to use more than 6 cubes of each color. Vasya finishes building the tower as soon as it contains 6 cubes of some two colors. How many different towers can Vasya build?

Answer 1

There are cubes of three colours. For convenience let the colours are red(R), blue(B) and green (G). Tower building will be finished as soon as 6 cubes of each of two colours will be used. These two colours may be red, blue or red, green or blue, green.

Case-1: The tower has 6 cubes of red and 6 cubes of blue colour and some of green colour

In this case following situations may occur.

6 red, 6 blue

6 red, 6 blue, 1 green

6 red, 6 blue, 2 green

6 red, 6 blue, 3 green

6 red, 6 blue, 4 green

6 red, 6 blue, 5 green

In every situation one cube of either red or blue is to be placed at the top of the tower to finish it. So the top most cube can be selected by 2 ways ( either red or blue) . This cube is kept aside. Remaining cubes are to be arranged as follows.

6 red, 6 blue cubes can be arranged by

$\frac{12!}{6!*6!} ways = 924 ways$

6 red, 6 blue, 1 green cubes can be arranged by $2*\frac{12!}{6!*5!} ways = 11088 ways$

6 red, 6 blue, 2 green cubes can be arranged by $2*\frac{13!}{6!*5!*2!} ways = 72072 ways$

6 red, 6 blue, 3 green cubes can be arranged by $2*\frac{14!}{6!*5!*3!} ways = 336336 ways$

6 red, 6 blue, 4 green cubes can be arranged by $2*\frac{15!}{6!*5!*4!} ways = 1261260 ways$

6 red, 6 blue, 5 green cubes can be arranged by $2*\frac{16!}{6!*5!*5!} ways = 4036032 ways$

Total number of ways of formation of tower containing 6red, 6 blue and green cubes less than 6 is

924+11088+72072+336336+1261260+4036032 = 5717712

Case-2

Similarly total number of ways of formation of tower containing 6 blue, 6 green and red cubes less than 6 is 5717712

Case-3

And total number of ways of formation of tower containing 6red, 6green and blue cubes less than 6 is 5717712

So Vasya can make towers in 3*5717712 = 17153136 ways.