Let A = {1, 2, 3, 4, 5} and B = {1, 2, 3, 4, 5, 6}. Then the number of functions f : A B satisfying f ( 1 ) + f ( 2 ) = f ( 4 ) - 1 is equal to __________ . [2023]

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Solution

Q.
Let A = {1, 2, 3, 4, 5} and B = {1, 2, 3, 4, 5, 6}. Then the number of functions $f : A \to B$ satisfying $f (1) + f (2) = f (4) - 1$ is equal to __________ . [2023]

Ans.
(360)

We have, $A = {1, 2, 3, 4, 5}; B = {1, 2, 3, 4, 5, 6}$

Also, $f (1) + f (2) = f (4) - 1 \Rightarrow f (1) + f (2) + 1 = f (4)$

Here, $f (4) \leq 6$

$f (1) + f (2) + 1 \leq 6$

$f (1) + f (2) \leq 5$

At   $f (1) = 1, then, f (2) = 1, 2, 3, 4 \Rightarrow 4 mappings,$

$f (1) = 2, then, f (2) = 1, 2, 3 \Rightarrow 3 mappings,$

   $f (1) = 3, then, f (2) = 1, 2 \Rightarrow 2 mappings$

   $f (1) = 4, then, f (2) = 1 \Rightarrow 1 mapping$

$f (3) and f (5) have 6 mappings each.$

$∴ Total number of functions = 10 \times 6 \times 6 = 360$

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