A) \[{{3}^{n}}-{{2}^{n}}\]
B) \[{{3}^{n}}-{{2}^{n}}-1\]
C) \[3\,({{2}^{n}}-1)\]
D) \[{{3}^{n}}-3\,({{2}^{n}}-1)\]
Correct Answer: D
Solution :
\[\because \] \[A=\{1,\,\,2,\,\,3,\,.....,\,n\}\] and \[B=\{a,\,b,\,c\}\] \[\therefore \] \[n(A)=n\] and \[n(B)=3\] Number of onto function from A to B \[=\underset{r\,=\,1}{\overset{3}{\mathop{\Sigma }}}\,{{(-1)}^{3-r}}{{\,}^{3}}{{C}_{r}}{{r}^{n}}\] \[={{(-1)}^{2}}{{\,}^{3}}{{C}_{1}}{{(1)}^{n}}+{{(-1)}^{1}}{{\,}^{3}}{{C}_{2}}{{(2)}^{n}}+{{(-1)}^{0}}{{\,}^{3}}{{C}_{3}}{{3}^{n}}\] \[={{3}^{n}}-3\cdot {{2}^{n}}+3\] \[={{3}^{n}}-3\,({{2}^{n}}-1)\]
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