Before You Begin

Question #40582

Difficulty: 8/10 | 🟢 Free

The Question

Let $S = \sum_{k=1}^{5} \frac{2k+1}{k^2(k+1)^2}$ and $n = \frac{1}{\sqrt{1 - S}}$ . Find the number of integer solutions $(x_1, x_2, x_3, x_4)$ to the equation $x_1 + x_2 + x_3 + x_4 = n$ such that $|x_i| \le 5$ for all $i = 1, 2, 3, 4$ .

Options

A.
600
B.
680
C.
540
D.
80

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Distributing Identical Items (Stars and Bars)|Method of Differences|Square Root and Modulus Logic

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of telescoping series (Method of Differences) with constrained integer partitions.

The primary trap lies in the Modulus Gatekeeper ( $|x_i| \le 5$ ).

Students often correctly shift the variables to non-negative integers ( $y_i = x_i + 5$ ) but forget to enforce the new upper bound ( $y_i \le 10$ ), erroneously calculating the unconstrained Stars and Bars result $\binom{17}{3} = 680$ .

Furthermore, the problem rewards advanced algebraic symmetry; recognizing that finding the coefficient of $t^{26}$ is identical to finding the coefficient of $t^{14}$ significantly reduces the Inclusion-Exclusion Principle (IEP) computational burden.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Permutations and Combinations - Class 11
Concept	Distributing Identical Items (Stars and Bars)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free