A) \[{{2}^{100}}-2\]
B) \[{{2}^{100}}\]
C) \[{{2}^{99}}-2\]
D) \[{{2}^{99}}\]
Correct Answer: A
Solution :
If A and B are two sets having m and n elements respectively such that \[1\le n\le m,\]then number of onto mappings from A to B \[=\sum\limits_{r=1}^{n}{{{(-1)}^{n-r}}}{{\,}^{n}}{{C}_{r}}{{r}^{m}}\] Here, \[m=100,n=2\] \[\therefore \] The number of onto mappings from A to B \[=\sum\limits_{r=1}^{2}{{{(-1)}^{2-r}}}{{\,}^{2}}{{C}_{r}}{{r}^{100}}\] \[={{(-1)}^{2-1}}.\,{{\,}^{2}}{{C}_{1}}{{.1}^{100}}+{{(-1)}^{2-2}}.{{\,}^{2}}{{C}_{2}}{{.2}^{100}}\] \[=-2+{{2}^{100}}={{2}^{100}}=2.\] Note: If set A has m elements and set B has n elements, then number of into functions from A to B is \[{{n}^{m}}.\]
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