Animated Solution for Mathematics - Binomial Theorem: If $a,b$ and $c$ are the greatest values of ${}^{19}{C}_p{,}^{20}{C}_q$ and ${}^{21}{C}_r$, respectively, then :

Welcome, future engineer! Today, we are not just solving a problem; we are exploring the elegant architecture of Pascal's Triangle. When you look at the sequence of binomial coefficients ${}^n{C}_r$ for a fixed $n$, you are looking at a discrete bell curve.

It rises, hits a peak, and falls with perfect symmetry. Our goal today is to find the relationship between the peaks of three consecutive rows: $n=19,20,$ and $21$.

Identifying the Peaks

First, let us locate our targets. We are looking for the greatest values, which we have labeled $a,b,$ and $c$.

For $n=19$, which is odd, the distribution has two central peaks. The middle terms occur at $r=\frac{19-1}{2}=9$ and $r=\frac{19+1}{2}=10$. Thus, our first value is:

Next, for $n=20$, an even number, we have a single, sharp peak at the center. The middle term is $r=\frac{20}{2}=10$. So, our second value is:

Finally, for $n=21$, another odd number, we return to the two-peak scenario. The middle terms are $r=\frac{21-1}{2}=10$ and $r=\frac{21+1}{2}=11$. Thus, our third value is:

The Time Machine (The Reduction Formula)

Now, we face a challenge. We have three values, but they belong to different 'generations' of binomial expansions. How do we compare them? We need a bridge. That bridge is the reduction formula:

Think of this formula as a time machine. It allows us to step back from a higher $n$ to a lower $n$. It is the key that unlocks the relationship between $a,b,$ and $c$.

Connecting the Dots

Let us connect $b$ to $a$. We know $b{=}^{20}{C}_{10}$. Applying our formula:

Since ${}^{19}{C}_9$ is exactly $a$, we get:

Now, let us connect $c$ to $b$. We know $c{=}^{21}{C}_{11}$. Applying the formula again:

Since ${}^{20}{C}_{10}$ is $b$, we get:

The Final Synthesis

We have two beautiful ratios: 1. $\frac{a}{1}=\frac{b}{2}$ 2. $\frac{b}{11}=\frac{c}{21}$

To unite them, we need the $b$ term to be identical in both. The least common multiple of $2$ and $11$ is $22$. Let us adjust our ratios:

Multiply the first ratio by $\frac{1}{11}$: $\frac{a}{11}=\frac{b}{22}$. Multiply the second ratio by $\frac{1}{2}$: $\frac{b}{22}=\frac{c}{42}$.

When we combine these, we arrive at the final, elegant relationship:

This result is not just a set of numbers; it is the geometric truth of how these binomial peaks grow relative to one another. You have successfully navigated the complexity of binomial distributions using nothing but logic and a powerful identity. Keep this clarity with you as you tackle the next problem!