JEE Main 2026 — Jan 21, Shift 1 — Mathematics solution

JEE Main2026Jan 21, Shift 1Mathematics

The number of strictly increasing functions f from the set ${1, 2, 3, 4, 5, 6}$ to the set $(1, 2, 3, \dots, 9)$ such that $f (i) \neq = i$ for $1 \leq i \leq 6$ , is equal to:

A
21
B
27
C
22
D
28

Solution

Since $f$ is strictly increasing, $f (1) \geq 2$ (because $f (1) \neq = 1$ ). Once $f (1)$ is chosen, the remaining 5 values are any 5-element strictly-increasing selection from the codomain values above $f (1)$ , and they must all exceed their indices to maintain $f (i) \neq = i$ (which here means $f (2) \geq 3$ , $f (3) \geq 4$ , etc., automatically satisfied if $f (1) \geq 2$ ).

Counting by $f (1)$ :

$f (1) = 2$ : choose 5 from ${3, 4, \dots, 9}$ (7 elements) $\Rightarrow (5 7) = 21$ .

$f (1) = 3$ : choose 5 from ${4, 5, \dots, 9}$ (6 elements) $\Rightarrow (5 6) = 6$ .

$f (1) = 4$ : choose 5 from ${5, 6, 7, 8, 9}$ $\Rightarrow (5 5) = 1$ .

Total $= 21 + 6 + 1 = 28$ .

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