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Let \(E=\{1,2,3,4\}\) and \(F=\{1,2\}\). Then the number of onto functions from \(\mathrm{E}\) to \(\mathrm{F}\) is
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The correct answer is:
14
If set \(A\) has \(m\) elements and set \(B\) has \(\mathrm{n}\) elements then number of onto functions from \(A\) to \(B\) is
\(\sum_{r=1}^n(-1)^{n-r}{ }^n C_r r^m \text { where } 1 \leq n \leq m\)
Here \(E=\{1,2,3,4\}, F=\{1,2\}\)
\(m=4, n=2\)
\(\therefore\) No. of onto functions from \(\mathrm{E}\) to \(\mathrm{F}\)
\(\begin{aligned}
& =\sum_{r=1}^2(-1)^{2-r ~2} C_r(r)^4 \\
& =(-1)^2 C_1+{ }^2 C_2(2)^4=-2+16=14
\end{aligned}\)
\(\sum_{r=1}^n(-1)^{n-r}{ }^n C_r r^m \text { where } 1 \leq n \leq m\)
Here \(E=\{1,2,3,4\}, F=\{1,2\}\)
\(m=4, n=2\)
\(\therefore\) No. of onto functions from \(\mathrm{E}\) to \(\mathrm{F}\)
\(\begin{aligned}
& =\sum_{r=1}^2(-1)^{2-r ~2} C_r(r)^4 \\
& =(-1)^2 C_1+{ }^2 C_2(2)^4=-2+16=14
\end{aligned}\)
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