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Question #41086

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all 5-tuples $(x_1, x_2, x_3, x_4, x_5)$ of non-negative integers such that $\sum_{i=1}^5 x_i = 25$ and $x_i \le 9$ for each $i = 1, 2, 3, 4, 5$ . Find the cardinality of $S$ .

Options

A.
5631
B.
3111
C.
4497
D.
5621

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Inclusion-Exclusion Principle|Multinomial Theorem|Properties of Binomial Coefficients

Why This Question Is Tricky

Common Mistakes

This problem tests the application of the Multinomial Theorem and Inclusion-Exclusion Principle via generating functions.

The specific trap lies in the binomial expansion of $(1-x^{10})^5$ .

Students frequently mishandle the alternating sign parity $(-1)^k$ of the coefficients (e.g., subtracting the $10x^{20}$ term instead of adding it, leading to 3111) or forget the binomial coefficients $\binom{5}{k}$ entirely (leading to 4497), resulting in incorrect linear combinations of the unconstrained solution spaces.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 11
Concept	Inclusion-Exclusion Principle
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free