Significance of Pascal’s Identity

We know the Pascal’s Identity very well, i.e. n_cr = n-1_cr + n-1_cr-1

A curious reader might have observed that Pascal’s Identity is instrumental in establishing recursive relation in solving binomial coefficients. It is quite easy to prove the above identity using simple algebra. Here I’m trying to explain it’s practical significance.

Recap from counting techniques, n_cr means selecting r elements from n elements. Let us pick a special element k from these n elements, we left with (n – 1) elements.

We can group these r elements selection n_cr into two categories,

1) group that contains the element k.

2) group that does not contain the element k.

Consider first group, the special element k is in all r selections. Since k is part of r elements, we need to choose (r – 1) elements from remaining (n – 1) elements, there are n-1_cr-1 ways.

Consider second group, the special element k is not there in all r selections, i.e. we will have to select all the r elements from available (n – 1) elements (as we must exclude element k from n). This can be done in n-1_cr ways.

Now it is evident that sum of these two is selecting r elements from n elements.

There can be many ways to prove the above fact. There might be many applications of Pascal’s Identity. Please share your knowledge.

— Venki.