If ²ⁿC₃ : ⁿC₃ = 10 : 1, then the ratio (n² + 3n) : (n² - 3n + 4) is

Q: Wait, why do we reject the negative root or other values? How do I know $n=8$ is the only answer?

Great question! In combinatorics, the 'n' in $^nC_r$ must be a non-negative integer, and specifically, $n \ge r$. Since we have $^nC_3$, we must have $n \ge 3$. Our algebraic simplification led to a linear equation, which naturally yields a unique solution. Always check your domain constraints!

Q: Can I just expand the combinations into polynomials first?

You could, but that is a trap! Expanding $(2n)(2n-1)(2n-2)$ into a cubic polynomial will make your algebra messy and prone to calculation errors. Always look for cancellations first—it is the hallmark of a smart JEE aspirant.

Question

If $^{2n}C_3 : ^nC_3 = 10 : 1$, then the ratio $(n^2 + 3n) : (n^2 - 3n + 4)$ is

Options:

(A) $35 : 16$

(B) $27 : 11$

(C) $65 : 37$

(D) $2 : 1$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

We are tasked with solving the combinatorics problem defined by the ratio:
$$^{2n}C_3 : ^nC_3 = 10 : 1$$

Our objective is to determine the value of $n$ and subsequently evaluate the ratio:
$$\frac{n^2 + 3n}{n^2 - 3n + 4}$$

### Decoding the Language of Combinations

To solve this, we utilize the standard combination formula:
$$^nC_r = \frac{n!}{r!(n-r)!}$$

Applying this to our given ratio, we obtain:
$$\frac{\frac{(2n)!}{3!(2n-3)!}}{\frac{n!}{3!(n-3)!}} = 10$$

Dividing by a fraction is equivalent to multiplying by its reciprocal. The $3!$ terms in the numerator and denominator cancel out, simplifying the expression to:
$$\frac{(2n)!}{(2n-3)!} 	imes \frac{(n-3)!}{n!} = 10$$

### The Factorial Expansion

We now expand the factorials to identify common terms for cancellation. We express the factorials as follows:
$$(2n)! = (2n)(2n-1)(2n-2)(2n-3)!$$
$$n! = n(n-1)(n-2)(n-3)!$$

Substituting these expansions back into our equation, the $(2n-3)!$ and $(n-3)!$ terms vanish. This leaves us with:
$$\frac{(2n)(2n-1)(2n-2)}{n(n-1)(n-2)} = 10$$

### The Algebraic Collapse

We factor $(2n-2)$ as $2(n-1)$ to simplify the expression further:
$$\frac{(2n)(2n-1) \cdot 2(n-1)}{n(n-1)(n-2)} = 10$$

Assuming $n \ge 3$, we cancel $n$ and $(n-1)$ from the numerator and denominator. This results in the linear equation:
$$\frac{4(2n-1)}{n-2} = 10$$

Cross-multiplying yields $8n - 4 = 10n - 20$. Solving for $n$, we find:
$$2n = 16 \Rightarrow n = 8$$

### Final Calculation

With $n = 8$ determined, we substitute this value into our target expression:
$$\frac{n^2 + 3n}{n^2 - 3n + 4}$$

Calculating the numerator:
$$8^2 + 3(8) = 64 + 24 = 88$$

Calculating the denominator:
$$8^2 - 3(8) + 4 = 64 - 24 + 4 = 44$$

The final ratio is:
$$\frac{88}{44} = 2$$

Thus, the final result is **$2 : 1$**.

Animated Solution for Mathematics - Permutations and Combinations: If $^{2 n} C_{3} :^{n} C_{3} = 10 : 1$ , then the ratio $(n^{2} + 3 n) : (n^{2} - 3 n + 4)$ is

Select Answer:

Visualized Solution

Analyze the Given Condition

Recall the Combination Formula

Expand the Combinations

Simplify the Complex Fraction

Expand Factorials to Cancel Terms

Cancel Common Factorial Terms

Factorize and Simplify Algebra

Solve the Linear Equation for n

Substitute n into the Target Expression

Final Ratio and Conclusion

The Sigma Insight: Combinations and Selection

Analyzing the Setup

Decoding the Language of Combinations

The Factorial Expansion

The Algebraic Collapse

Final Calculation

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Animated Solution for Mathematics - Permutations and Combinations: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If 2nC3​:nC3​=10:1, then the ratio (n2+3n):(n2−3n+4) is