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Question #40581

Difficulty: 8/10 | 🟢 Free

The Question

A variable plane intersects the $x, y,$ and $z$ axes at exactly one point each, denoted by $A(a, 0, 0)$ , $B(0, b, 0)$ , and $C(0, 0, c)$ respectively, where $a, b,$ and $c$ are non-negative integers satisfying $a + b + c = 20$ . Let $N$ be the number of such distinct planes. If $V$ is the maximum volume of the tetrahedron $OABC$ (where $O$ is the origin), find the value of $N + V$ .

Options

A.
220
B.
280
C.
277
D.
231

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Distributing Identical Items (Stars and Bars)|Variable plane intercepts|Collinearity vs Coplanarity

Why This Question Is Tricky

Common Mistakes

This question tests the geometric validity of algebraic solutions in 3D geometry, specifically targeting the 'Collinearity vs Coplanarity' and degenerate plane traps.

Students often blindly apply the 'Stars and Bars' formula for non-negative integer solutions to $a+b+c=20$ , yielding $\binom{22}{2} = 231$ .

However, they fail to realize that if a plane intersects an axis at the origin ( $a=0$ ), it must pass through the origin.

For the plane to intersect the other axes at exactly one point (rather than containing the entire axis), those intercepts must also be zero ( $b=c=0$ ), which violates the sum constraint $a+b+c=20$ .

Thus, the intercepts must be strictly positive integers, requiring the formula for positive solutions $\binom{19}{2} = 171$ .

This acts as a rigorous filter against superficial substitution.

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