CAT 2024 Slot 1 QA Question & Solution
Question
Consider two sets $A = \left\{2, 3, 5, 7, 11, 13 \right\}$ and $B = \left\{1, 8, 27 \right\}$. Let f be a function from A to B such that for every element in B, there is at least one element a in A such that $f(a) = b$. Then, the total number of such functions f is
Options
Solution
$Set\ A={2,3,5,7,11,13}$, so $|A| = 6$
$Set\ B={1, 8, 27}$, so $|B|=3$
Without any restrictions, each element in A can map to any of the 3 elements in B. Thus, the total number of functions is: $3^6=729$
Excluding Functions That Miss One Element in B: If a function does not map to an element in B, there are 2 elements in B left for mapping. The total number of such functions (for each specific element not mapped) is: $2^6=64$
Since there are 3 elements in B, the total number of such functions is: $3\times64=192$
Adding Back Functions That Miss Two Elements in B:
If a function misses two elements in B, there is only 1 element left for mapping. The total number of such functions is: $1^6=1$.
Since there are $^3C_2$ ways to choose which two elements are missed, the total number of such functions is: 3
Using the inclusion-exclusion principle, the number of functions where all elements of B are mapped by at least one element of A is:
$729-192+3=540$