Let R = {a, b, c, d, e} and S = {1, 2, 3, 4}. Total number of onto functions f : R S such that f ( a ) 1 , is equal to ________ . [2023]

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Solution

Q.
Let R = {a, b, c, d, e} and S = {1, 2, 3, 4}. Total number of onto functions $f : R \to S$ such that $f (a) \neq 1,$ is equal to ________ . [2023]

Ans.
(180)

Total onto functions

$= 4^{5} - {}^{4}{C_{1}} \times 3^{5} + {}^{4}{C_{2}} \times 2^{5} - {}^{4}{C_{3}} \times 1^{5} = 240$

Now, when $f (a) = 1$

$⌊ 4 + \frac{⌊ 4}{⌊ 2 ⌊ 2} \times ⌊ 3 = 24 + 36 = 60$

So, required number of onto functions $= 240 - 60 = 180$

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