The number of functions f, from the set $A=\left\{x \in N: x^{2}-10 x+9 \leq 0\right\}$ to the set $B=\left\{n^{2}: n \in N\right\}$ such that $f(x) \leq(x-3)^{2}+1$, for every $x \in A$, is ______.

JEE Main · 2022 · 27 Jul 2022 · Shift 2 · Sets Relations and Functions

The number of functions f, from the set

A = {x \in N : x^{2} - 10 x + 9 \leq 0}

to the set

B = {n^{2} : n \in N}

such that

f (x) \leq (x - 3)^{2} + 1

, for every

x \in A

, is ______.

Answer: 1440

From

x^{2} - 10 x + 9 \leq 0

, we get

A = {1, 2, 3, 4, 5, 6, 7, 8, 9}

. For each

x \in A

, we count the number of possible values in

B

satisfying

f (x) \leq (x - 3)^{2} + 1

x = 1

: 2 choices;

x = 2

: 1 choice;

x = 3

: 1 choice;

x = 4

: 1 choice;

x = 5

: 2 choices;

x = 6

: 3 choices;

x = 7

: 4 choices;

x = 8

: 5 choices;

x = 9

: 6 choices.
Total functions =

2 \times 1 \times 1 \times 1 \times 2 \times 3 \times 4 \times 5 \times 6 = 2 \times 6! = 1440

JEE Main · 2022 · 27 Jul 2022 · Shift 2 · Integer answer