Sets, Relations & Functions
Functions
nta_abhyas_2025
Grade 11
Question:
Total number of functions = $3^5$. Since each of 1, 2, 3, 4, or 5 can correspond to any of $a$, $b$, or $c$. The number of functions that corresponds to only one element of $B$ is $^3C_1 imes 1^3$ and the number of functions that correspond to almost two elements of $B$ is $^3C_2 imes 2^5$. Total number of onto functions = $3^5 - ^3C_1 imes 1^3 - (^3C_2 imes 2^5)$ (using $^3C_1 imes 1^3$ repeated twice in $^3C_2 imes 2^5$). What is the result?
Step-by-Step Solution
Key Concept: Inclusion-exclusion principle applied to surjective (onto) functions counts total functions minus those missing at least one element.
Using inclusion-exclusion principle: Total functions = $3^5 = 243$. Functions mapping to exactly one element = $^3C_1 imes 1^5 = 3$. Functions mapping to at most two elements = $^3C_2 imes 2^5 = 3 imes 32 = 96$. Using inclusion-exclusion, onto functions = $243 - 3 - 96 = 150$. Alternatively, $243 - 3(1) - 3(32) + 3(1) = 243 - 3 - 96 + 3 = 150$.
Correct Answer: 150