Let A = {1, 2, 3, 4} and B = {1, 4, 9, 16}. Then the number of many-one functions f : A B such that is equal to : [2025]

Question

Let A = {1, 2, 3, 4} and B = {1, 4, 9, 16}. Then the number of many-one functions f : A $\to$ B such that $I \in f (A)$ is equal to : [2025]

Smart Achievers · Accepted Answer

(3)

Here, n(A) = 4, n(B) = 4

Total number of functions from A to B = $4^{4}$ = 256

Number of one-one functions from A to B = ${}^{4}P_{4}$ = 4! = 24

Number of many-one functions from A to B = 256 – 24 = 232

Number of many-one function for which $1 \notin f (A) = 3^{4}$ = 81

$∴$ Required number of many-one functions = 232 – 81 = 151.

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