
We have a set of seven colors  Vibgyor. Now the question is how many
combinations (subsets) of 4, 3, 2, and 1 colors can be selected from the
7color set? Suppose, at this stage the order of the numbers in the subset
doesn't matter. It means the subset of four colors "bluegreenredyellow"
is considered to be the same as the subset "greenblueyellowred."
There is a
formula
how to figure out the amount of unordered combinations of X with Y
numbers. It is equal to Y!/(X! (YX)!). X! is factorial of X, Y! is
factorial of Y and (YX)! is factorial of (YX). Any factorial is a
product of consecutive numbers starting from 1 till the final number
inclusive. For example, 4! equal to 1x2x3x4 or 24.
Thus, the amount of unordered combinations of X with Y is:
1) 4 numbers: 7!/(4! x (74)!) = 5040/(24 x 6) = 35;
2) 3 numbers: 7!/(3! x (73)!) = 5040/(6 x 4) = 35;
3) 2 numbers: 7!/(2! x (72)!) = 5040/(2 x 120) = 21;
1) 1 number: 7!/(1! x (71)!) = 5040/(1 x 720) = 7.
As you can see, because of the formula, the amount of 3color combinations
in a 7color set is the same as the amount of 4color combinations, i.e.
35.
Now as we have figured out the total amount of all possible unordered
combinations the question is: what are unique schemes in which these
combinations can be applied to paint the pyramid? It will be convenient to
imagine we are painting our pyramids on the flat cardboard, as in the
diagrams, before folding up.
If we take the 4color subset (say, blue, green, red, and yellow), it can
be applied in only 2 distinctive ways, as shown in Figs. 1 and 2. Any
other way will only result in one of these when the pyramids are folded
up. If we take three colors (say, green, red, and yellow), they may be
applied in only 3 was shown in Figs. 3, 4, and 5. Two colors (say, green
and yellow) may be also applied only in 3 ways shown in Figs. 6, 7, and 8.
Any single color (say, blue) may obviously be applied in only 1 way shown
in Fig. 9.
Multiplying the number of unordered combinations by the number of
distinctive colored scheme for that combination we obtain the following
amounts:
1) 4 colors: 2 x 35 = 70;
2) 3 colors: 3 x 35 = 105;
3) 2 colors: 3 x 21 = 63;
4) 1 color: 1 x 7 = 7.
Thus, a total amount of the unique ways how the pyramid me be painted in,
using each time the colors from the solar spectrum is the sum of these
four numbers: 70 + 105 + 63 + 7 = 245. 
