Best Way of Solving the 12 Days of Christmas Problem

I have mentioned in my last journal that one of the approaches of calculating the total amount of presents in the 12 days of Christmas song is to use Pascal's triangle, but that could be tedious for more challenging situations.  Knowing that the binomial coefficients are "specific addresses" in Pascal's triangle as shown in the figure, it becomes way easier to solve problems similar but more complicated than the 12 days of Christmas. 

Here is how it works:

The binomial coefficients are written in any form similar to "n choose k".  That refers to the kth element in the nth row in Pascal's triangle.  To verify, one can use the formula first and then compare with the corresponding value in Pascal's trianlge.

So if we, for example, are trying to choose two items out of seven options, the answer according to the formula is:

and sure enough, if we look at the 2nd element in the 7th row in Pascal's triangle (keeping in mind that the first row in Pascal's triangle is the 0th row), the value is 21.

Hence, the nice relationship between the binomial coefficients and Pascal's triangle helps a lot in solving more complex problems.