Key Concepts and Formulas

• Permutation Formula: ${ }^{n} \mathrm{P}_{r} = \frac{n!}{(n-r)!}$
• Factorial definition: $n! = n \cdot (n-1) \cdot (n-2) \cdot ... \cdot 2 \cdot 1$

Step-by-Step Solution

Step 1: Write down the given ratio of permutations and apply the permutation formula. We are given ${ }^{2 n+1} \mathrm{P}_{n-1}:{ }^{2 n-1} \mathrm{P}_{n}=11: 21$. Using the permutation formula, we can write this as: $\frac{{ }^{2 n+1} \mathrm{P}_{n-1}}{{ }^{2 n-1} \mathrm{P}_{n}} = \frac{\frac{(2n+1)!}{(2n+1-(n-1))!}}{\frac{(2n-1)!}{(2n-1-n)!}} = \frac{11}{21}$ Simplifying the denominators: $\frac{\frac{(2n+1)!}{(n+2)!}}{\frac{(2n-1)!}{(n-1)!}} = \frac{11}{21}$

Step 2: Simplify the expression by rewriting the factorials. We rewrite the factorials to simplify the expression: $\frac{(2n+1)!}{(n+2)!} \cdot \frac{(n-1)!}{(2n-1)!} = \frac{11}{21}$ $\frac{(2n+1)(2n)(2n-1)!}{(n+2)(n+1)(n)(n-1)!} \cdot \frac{(n-1)!}{(2n-1)!} = \frac{11}{21}$ Canceling out the common terms $(2n-1)!$ and $(n-1)!$, we get: $\frac{(2n+1)(2n)}{(n+2)(n+1)(n)} = \frac{11}{21}$

Step 3: Further simplification $\frac{2(2n+1)}{(n+2)(n+1)} = \frac{11}{21}$

Step 4: Cross-multiply and form a quadratic equation. Cross-multiplying, we have: $21 \cdot 2(2n+1) = 11(n+2)(n+1)$ $42(2n+1) = 11(n^2 + 3n + 2)$ $84n + 42 = 11n^2 + 33n + 22$ $0 = 11n^2 + 33n - 84n + 22 - 42$ $11n^2 - 51n - 20 = 0$

Step 5: Solve the quadratic equation. We can factor the quadratic equation as: $11n^2 - 55n + 4n - 20 = 0$ $11n(n - 5) + 4(n - 5) = 0$ $(11n + 4)(n - 5) = 0$ This gives us two possible solutions for $n$: $n = 5 \quad \text{or} \quad n = -\frac{4}{11}$

Step 6: Determine the valid solution for $n$. Since $n$ must be a positive integer (because of the permutation ${ }^{2 n-1} \mathrm{P}_{n}$), we discard the negative solution $n = -\frac{4}{11}$. Therefore, $n = 5$.

Step 7: Calculate $n^2 + n + 15$. Now, we substitute $n = 5$ into the expression $n^2 + n + 15$: $n^2 + n + 15 = (5)^2 + 5 + 15 = 25 + 5 + 15 = 45$

Common Mistakes & Tips

• Remember that $n$ in ${ }^{n} \mathrm{P}_{r}$ must be a non-negative integer. Also, $n \geq r$.
• Be careful with factorial simplification; double-check which terms can be canceled.
• Always check if the solution obtained makes sense in the context of the problem.

Summary We are given a ratio of permutations and asked to find the value of $n^2 + n + 15$. We used the permutation formula to express the given ratio in terms of factorials, simplified the expression, and obtained a quadratic equation. Solving the quadratic equation, we found two possible values for $n$, but only the positive integer value $n=5$ is valid. Substituting $n=5$ into $n^2 + n + 15$, we get the final answer of 45.

Final Answer The final answer is \boxed{45}.