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GO Classes DPP | GATE CS | Discrete Mathematics| Functions

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The total number of functions $f:{1,2,3,\ldots,12}\to{0,1}$ such that $f$ assigns $1$ to exactly two even integers less than $12$ and assigns $0$ to exactly three odd integers greater than $3$ is$\_\_\_\_\_\_.$

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The even integers less than $12$ in the domain are $2,4,6,8,10$. There are $5$ such integers.

Exactly two of them must be assigned $1$, so this can be done in $\binom{5}{2}=10$ ways.

The odd integers greater than $3$ in the domain are $5,7,9,11$. There are $4$ such integers.

Exactly three of them must be assigned $0$, so this can be done in $\binom{4}{3}=4$ ways.

The remaining unrestricted elements are $1,3,12$. Each of these can be assigned either $0$ or $1$, so there are $2^3=8$ ways.

Hence the total number of functions is $\binom{5}{2}\binom{4}{3}2^3=10\cdot 4\cdot 8=320$.

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