Animated Solution for Mathematics - Binomial Theorem: ${\sum }_{k=0}^6{}^{51-k}{C}_3$ is equal to

Welcome, fellow traveler on the road to JEE excellence. Today, we aren't just solving a summation; we are witnessing a mathematical domino effect.

When you look at the expression ${\sum }_{k=0}^6{}^{51-k}{C}_3$, it might look like a daunting list of combinations. You might be tempted to reach for your calculator, but I want you to pause. In the world of competitive mathematics, we don't calculate; we observe, we manipulate, and we conquer.

The Visual Awakening

Let us expand the summation. By plugging in $k=0,1,2,3,4,5,6$, we get the sequence:

It looks a bit messy, doesn't it? Let's flip it around, as there is immense power in order. Let us write it in ascending order of the upper index:

Now, look closely. The lower index is stuck at $3$, while the upper index is climbing the ladder. This is the signature of Pascal's Identity:

To use the identity, we need a partner with a lower index of $4$.

The Add-and-Subtract Trick

This is where the magic happens. We need a ${}^{45}{C}_4$ to start the chain reaction. So, we perform a classic maneuver: we add it, and we subtract it.

By adding ${}^{45}{C}_4$ to the start of our series, we haven't changed the value, but we have unlocked the potential for a collapse:

The Domino Effect

Look at the first parenthesis. According to Pascal's Identity, ${}^{45}{C}_4+{}^{45}{C}_3$ becomes ${}^{46}{C}_4$. Now, our expression looks like this:

Do you see it? The ${}^{46}{C}_4$ we just created is now perfectly positioned to merge with the next term, ${}^{46}{C}_3$. They combine to form ${}^{47}{C}_4$.

This is the 'Hockey-Stick' effect in action. The merge continues, climbing the upper index one step at a time, like a wave moving through a crowd.

The Final Victory

As the wave reaches the end of our series, the final merge is ${}^{51}{C}_4+{}^{51}{C}_3$, which yields ${}^{52}{C}_4$. We are left with the result of this beautiful, cascading collapse, minus the term we introduced at the very beginning.

Our final answer stands elegant and clear:

Isn't that satisfying? We didn't need to compute a single large factorial. We simply understood the structure of the binomial coefficients and let the identity do the heavy lifting.

Remember this moment the next time you face a long summation. Don't calculate—look for the pattern, seed the identity, and watch the dominoes fall.