# Significance of Pascal’s Identity

We know the Pascal’s Identity very well, i.e. **n _{c}_{r} = n-1_{c}_{r} + n-1_{c}_{r-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 _{c}_{r} **means selecting

**elements from**

*r***elements. Let us pick a special element**

*n***from these**

*k***elements, we left with**

*n***(**elements.

*n – 1*)We can group these ** r** elements selection

**n**into two categories,

_{c}_{r}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

**selections. Since**

*r***is part of**

*k***elements, we need to choose (**

*r***) elements from remaining (**

*r – 1***) elements, there are**

*n – 1***n-1**

_{c}_{r-1}ways.

Consider second group, the special element ** k** is not there in all

**selections, i.e. we will have to select all the**

*r***elements from available (**

*r***) elements (as we must exclude element**

*n – 1***from**

*k***). This can be done in**

*n***n-1**ways.

_{c}_{r}Now it is evident that sum of these two is selecting ** r** elements from

**elements.**

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

— **Venki**. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

## Recommended Posts:

- Cassini’s Identity
- Program for Identity Matrix
- Euler's Four Square Identity
- Brahmagupta Fibonacci Identity
- Proizvolov's Identity
- Right most non-zero digit in multiplication of array elements
- Find the remainder when First digit of a number is divided by its Last digit
- Find the remaining balance after the transaction
- Count of integers that divide all the elements of the given array
- Count number of ways to get Odd Sum
- Percentage increase in the volume of cuboid if length, breadth and height are increased by fixed percentages
- Count the number of occurrences of a particular digit in a number
- Find number of factors of N when location of its two factors whose product is N is given
- Square free semiprimes in a given range using C++ STL