## Color printing

A color ink jet printer can place up to 8 drops at a single point. A drop may have a cyan (C), magenta (M), yellow (Y) or black (K) color. How many distinct shades may we achive at a single point, if we assume that mixing three drops of various (CMY) colors has the same effect as two black drops? (For example, the shade 3C+2Y+M+K is the same as 2C+Y+3K.)

• #### Hint

Simplify the situation by assuming that there are always eight drops.

• #### Solution

Imagine that if there are fewer than 8 drops, we will add drops of a clear color to bring the total count to eight. Then the total number of possible ways to print on a single point is the same as the ways to divide eight drops into five pigeonholes, i.e. $$\binom{12}4$$.

Of course, the shades which we can construct in various ways will be counted more than once – some twice, others three times. These are the cases which allow a color to be expressed as a combination of drops containing at least one C, at least one M and at least one Y.

So we will subtract the cases in which the colors CMY appear at least once, i.e. $$\binom{9}4$$ possibilities. Notice that these possibilities include even the cases where the colors CMY appear at least twice (e.g. 3C+2Y+2M is subtracted along with 2C+Y+M+2K and so this shade will be achievable only as C+4K).

We have used the fact that it is always possible to replace colored drops with a black drop, which of course is not true in reverse.

In the given situation it is possible to generate $$\binom{12}4-\binom{9}4=495-126=369$$ colored shades.  