Animated Solution for Mathematics - Binomial Theorem: The value of \sum_{r=0}^{20} ^{50-r}C_6 is equal to:

Welcome, fellow traveler in the world of mathematics. Today, we are going to unravel a problem that might look like a daunting wall of symbols, but is actually a beautiful, rhythmic dance of numbers.

We are looking at the summation ${\sum }_{r=0}^{20}{}^{50-r}{C}_6$. At first glance, it is just a series of binomial coefficients, but let us peel back the layers.

The Expansion

Let us start by simply writing out what this summation actually means. When we plug in $r=0$, we get ${}^{50}{C}_6$. When $r=1$, we get ${}^{49}{C}_6$.

This continues all the way down to $r=20$, which gives us ${}^{30}{C}_6$. So, our series is:

Notice how the upper index is decreasing? It is a bit like walking down a staircase.

The Reversal

To make this easier to visualize, let us reverse the order. It is much more natural to think about climbing up a staircase than walking down one.

Let us write it from the smallest upper index to the largest:

Now, we have a clear, ascending sequence. This is much better, isn't it?

The Logic Bridge

Now, how do we sum these? We need a tool. Enter the legendary Pascal's Identity:

This identity is the key to unlocking the entire problem. It tells us that if we have two binomial coefficients with the same upper index and consecutive lower indices, they merge into a single term with a higher upper index.

But look at our series: we have ${}^{30}{C}_6,{}^{31}{C}_6,\dots$. The upper indices are changing, and the lower index is fixed at $6$. We need to create the conditions for Pascal's Identity to work.

The Magic Trick

This is where we perform a little mathematical alchemy. We have ${}^{30}{C}_6$. To use Pascal's Identity, we desperately need a ${}^{30}{C}_7$.

But we don't have one! So, we do what any clever mathematician would do: we add it and subtract it. We add ${}^{30}{C}_7$ to the start of our series, and subtract it at the end to keep everything balanced.

Our series becomes:

The Domino Effect

Now, watch the magic. The first two terms, ${}^{30}{C}_7+{}^{30}{C}_6$, merge into ${}^{31}{C}_7$.

Now our series looks like:

But wait! Now we have ${}^{31}{C}_7+{}^{31}{C}_6$. They merge into ${}^{32}{C}_7$. This is a beautiful, cascading domino effect. Each merge creates a new term that perfectly pairs with the next term in the series.

This chain reaction continues all the way to the end. The final merge will be ${}^{50}{C}_7+{}^{50}{C}_6$, which gives us ${}^{51}{C}_7$.

And don't forget that ${}^{30}{C}_7$ we subtracted at the beginning! It is still waiting for us. So, our entire massive summation collapses into just two terms:

Isn't that elegant? We took a complex sum and reduced it to a simple subtraction. Keep this trick in your toolkit—it is a powerful way to handle telescoping sums in combinatorics.