JEE Advanced · Mathematics · 11. Sets & Relations

If \(n(X)={ }^m C_6\), then the value of \(m\) is

A 20
B 25
C 30
D 35

Verified Solution

Answer & Solution

Correct Answer

(A) 20

Step-by-step Solution

Detailed explanation

\(S=\{1,2,3,4,5,6\} \quad R: S \rightarrow S\)
Number of elements in \(R=6\)
and for each \((a, b) \in R ;|a-b| \geq 2\)
\(X \rightarrow\) set of all relation \(R: S \rightarrow S\)
If \(a=1, b=3,4,5,6 \rightarrow(4)\)
\(a=2, b=4,5,6 \rightarrow(3)\)
\(a=3, b=1,5,6 \rightarrow(3)\)
\(a=4, b=1,2,6 \rightarrow(3)\)
\(a=5, b=1,2,3 \rightarrow(3)\)
\(a=6, b=1,2,3,4 \rightarrow(4)\)
Total number of ordered pairs \((a, b)\)
s.t. \(|a-b| \geq 2 =20\)
\(\therefore n(X)=\) number of elements in \(\mathrm{X}={ }^{20} C_6\)
\(\therefore m=20\)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

Same subject

Let \(z_k=\cos \left(\frac{2 k \pi}{10}\right)+i \sin \left(\frac{2 k \pi}{10}\right) ; k=1,2, \ldots, 9\)

List - I	List - II
(A) For each \(z_k\) there exists a \(z_j\) such \(z_k \cdot z_j=1\)	(P) True
(B) There exists a \(k \in\{1,2, \ldots, 9\}\) such that \(z_1 \cdot z=z_k\) has no solution z in the set of complex numbers	(Q) False
(C) \(\frac{\left\|1-z_1\right\|\left\|1-z_2\right\|\ldots\left\|1-z_9\right\|}{10}\) equals	(R) 1
(D) \(1-\sum_{k=1}^9 \cos \left(\frac{2 k \pi}{10}\right)\) equals	(S) 2

JEE Advanced 2014 Hard

Explore more questions on app

From JEE Advanced

Explore more questions on app