JEE Main 2025 Apr 7 Shift 1, Mathematics Q25: Combinations

JEE Main2025Apr 7, Shift 1Mathematics

For $n \geq 2$ , let $S_{n}$ denote the set of all subsets of ${1, 2, \dots, n}$ with no two consecutive numbers. For example ${1, 3, 5} \in S_{6}$ , but ${1, 2, 4} \in / S_{6}$ . Then $n (S_{5})$ is equal to _____

Your answer

Solution

Count subsets of ${1, 2, 3, 4, 5}$ with no two consecutive elements, grouped by subset size.

Formula: the number of $r$ -element subsets of ${1, 2, \dots, n}$ with no two consecutive elements is $(r n - r + 1)$ .

For $n = 5$ , sum over $r = 0, 1, 2, 3$ :

$r$	$(r 6 - r)$	Count
$0$	$(0 6)$	$1$
$1$	$(1 5)$	$5$
$2$	$(2 4)$	$6$
$3$	$(3 3)$	$1$

Total: $1 + 5 + 6 + 1 = 13$ .

Sizes 4 and 5 are impossible without consecutivity in a set of 5 elements.

$∴ n (S_{5}) = 13$ .

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JEE Main 2025 Apr 7 Shift 1, Mathematics Q25: Combinations

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