UPSC NDA I 2024 Mathematics Paper

Concept:The number of onto functions from a set $A$ with $m$ elements to a set $B$ with $n$ elements ($m \ge n$) is given by the inclusion-exclusion principle: $n^m - \binom{n}{1}(n-1)^m + \binom{n}{2}(n-2)^m - \dots + (-1)^{n-1}\binom{n}{n-1}1^m$.Explanation:Here $A = \{1,2,3,4,5\}$ so $m=5$, and $B = \{6,7\}$ so $n=2$.An onto function requires every element of $B$ to have at least one preimage in $A$.Total possible functions from $A$ to $B$ are $2^5 = 32$.From these, subtract the functions that are not onto, i.e., those that miss at least one element of $B$.Number of functions that miss a specific element (say 6) is $(2-1)^5 = 1^5 = 1$ (all elements of $A$ map to the other element).There are $\binom{2}{1}=2$ ways to choose which element is missing.Thus, number of onto functions = $32 - 2 \times 1 = 30$.So the correct answer is $30$.Answer:$30$ (Option C)