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JEE Advanced · 2024 · 26 May 2024 · Paper 2 · Permutations and Combinations

Let

S = {1, 2, 3, 4, 5, 6}

and

X

be the set of all relations

R

from

S

to

S

that satisfy both the following properties.
i.

R

has exactly 6 elements.
ii. For each

(a, b) \in R

, we have

∣ a - b ∣ \geq 2

.
Let

Y = {R \in X

: The range of

R

has exactly one element

}

and

Z = {R \in X : R

is a function from

S

to

S

.

}

Let

n (A)

denote the number of elements in a set

A

.

If

n (X) =^{m} C_{6}

, then the value of m is ____.

Answer: 20

Solution

For each

a \in S

, we count the possible values of

b

such that

∣ a - b ∣ \geq 2

:

$a = 1$ : $b \in {3, 4, 5, 6}$ → 4 choices
$a = 2$ : $b \in {4, 5, 6}$ → 3 choices
$a = 3$ : $b \in {1, 5, 6}$ → 3 choices
$a = 4$ : $b \in {1, 2, 6}$ → 3 choices
$a = 5$ : $b \in {1, 2, 3}$ → 3 choices
$a = 6$ : $b \in {1, 2, 3, 4}$ → 4 choices

Total ordered pairs:

4 + 3 + 3 + 3 + 3 + 4 = 20

Since

R

has exactly 6 elements chosen from these 20 pairs:

n (X) = (6 20) =^{20} C_{6}

Therefore,

m = 20

.

JEE Advanced · 2024 · 26 May 2024 · Paper 2 · Numerical answer

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More JEE Advanced questions on Permutations and Combinations