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

JEE Main · 2026 · 21 Jan 2026 · Shift 1 · Permutations and Combinations

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 :

Answer: (D)

A strictly increasing map is determined by its image, a

6

-element subset of

{1, \dots, 9}

taken in increasing order, so

f (i) \geq i

holds automatically. The condition

f (i) \neq = i

then means

f (i) \geq i + 1

for every

i

. Writing

u_{i} = f (i) - i

, the

u_{i}

are non-decreasing with

1 \leq u_{i} \leq 3

, so their number equals the count of non-decreasing sequences of length

6

from

{1, 2, 3}

, namely

(2 6 + 2) = 28

JEE Main · 2026 · 21 Jan 2026 · Shift 1 · Single correct option