Before You Begin

Question #41007

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all ordered triplets $(x,y,z)$ of non-negative integers such that $x+y+z \le 20$ . The sample space $\Omega$ is defined as the subset of $S$ for which the expression $\frac{1}{x-y}$ is a well-defined real number. If a triplet is chosen uniformly at random from $\Omega$ , what is the probability that $z > x + y$ ?

Options

A.
$\frac{38}{165}$
B.
$\frac{380}{1771}$
C.
$\frac{4}{15}$
D.
$\frac{40}{161}$

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

Axiomatic Probability Constraints|Division into Groups|Unbounded Region Trap|Domain Restrictions

Why This Question Is Tricky

Common Mistakes

The question tests the 'Unbounded Region Trap' disguised within a discrete partition problem.

Students often misinterpret inequality constraints as an infinite sample space or fail to properly bound the summation.

Furthermore, the domain restriction $1/(x-y)$ acts as a hidden constraint that must be excluded from both the universal sample space and the favorable event space.

A common mistake is calculating the probability over the unrestricted partition space $\binom{23}{3}$ , leading to the trap answer $40/161$ , or forgetting to exclude the $x=y$ cases from the favorable events, leading to $4/15$ .

Quick Facts About This Question

Question Information

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