AP and TS EAMCET Engineering Model Paper 3

A one-one (injective) function maps distinct elements of $A$ to distinct elements of $B$.So we must assign to the $3$ elements of $A$ an ordered selection of $3$ distinct elements out of the $5$ elements of $B$.The first element of $A$ has $5$ choices, the second $4$ choices, and the third $3$ choices.Hence the number of one-one functions is $5 \times 4 \times 3 = {}^{5}P_{3} = 60$.In general, if $n(A)=r$ and $n(B)=n$ with $r \le n$, the number of one-one functions is ${}^{n}P_{r}$.Here $n=5$ and $r=3$, so the answer is ${}^{5}P_{3}$.Note that ${}^{5}C_{3}$ only counts unordered selections (subsets), which would not distinguish the different assignments to the elements of $A$.Therefore the correct option is ${}^{5}P_{3}$, i.e. option A.