PEA 2022 PEA Q23 | ISI MSQE

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2022 PEA Q23

$f : A \to B$ with $|A|=5$ , $|B|=2$ . How many onto functions?

Reveal answer and solution

Answer

Solution

1
Total functions from $A$ to $B$ : $2^5 = 32$ . Subtract the $2$ functions that are not onto
2
(constant maps to $\{1\}$ or to $\{2\}$ ): $32 - 2 = 30$ .

Answer structure / marking notes

By inclusion--exclusion, the number of onto maps is $\sum_{k=0}^{|B|}(-1)^k \binom{|B|}{k}(|B|-k)^{|A|}$ .

Content note

Imported from public/resources/isi/msqe/solutions/pea/2022/ISI_MSQE_PEA_2022_Solutions.tex. Question wording is retained from the available local TeX source; incomplete option blocks or ambiguous source status are flagged for review.