Animated Solution for Mathematics - Permutations and Combinations: If $\frac{{}^{2n+1}{P}_{n-1}}{{}^{2n-1}{P}_n}=11:21$, then $n^2+n+15$ is equal to:

Welcome, future engineer! Today, we are not just solving a math problem; we are embarking on a journey through the elegant world of permutations.

You have been presented with the ratio:

It looks intimidating, but in the JEE Advanced arena, complexity is often just a mask for a beautiful, hidden simplicity. Our mission is to unmask it.

Decoding the Permutation

First, let us ground ourselves in the fundamental definition. The permutation formula is our North Star:

When we apply this to our numerator, ${}^{2n+1}{P}_{n-1}$, we get:

Now, let us turn our attention to the denominator, ${}^{2n-1}{P}_n$. Applying the same logic:

The Art of Factorial Cancellation

Now, we assemble our pieces:

When we divide by a fraction, we multiply by its reciprocal:

This is where the magic happens. We use the "peeling" technique to expand the larger factorials until they match the smaller ones:

Substituting these back into our equation, the $(2n-1)!$ and $(n-1)!$ terms cancel out perfectly:

The Quadratic Bridge

The $n$ in the numerator and the $n$ in the denominator cancel out (given $n eq 0$), leaving us with:

Cross-multiplying gives us:

Rearranging everything to one side, we arrive at the quadratic equation:

Solving this via the middle-term split:

This yields two roots: $n=5$ and $n=-\frac{4}{11}$. Since $n$ must be a positive integer, we discard the negative fraction and conclude that $n=5$.

The final step is to evaluate $n^2+n+15$. Substituting $n=5$:

You have navigated the permutations, mastered the factorials, and solved the quadratic. The final answer is 45.