A) \[\frac{({{3}^{n}}{{.}^{6}}{{C}_{3}})}{{{6}^{n}}}\]
B) \[\frac{{{({{3}^{n}}-{{3.2}^{n}}+3)}^{6}}{{C}_{3}}}{{{6}^{n}}}\]
C) \[\frac{{{({{3}^{n}}-3)}^{6}}{{C}_{3}}}{{{6}^{n}}}\]
D) None of these
Correct Answer: B
Solution :
Let us define a onto function f from \[A:[{{r}_{1}},{{r}_{2}}.....,{{r}_{n}}]\] to B :[1, 2, 3] where \[{{r}_{1}},{{r}_{2}}\],......\[{{r}_{n}}\] are the readings of the n throws and 1, 2 and 3'are the numbers that appear in the n throws. Number of such functions \[M=N-[n(1)-n(2)+n(3)]\] Where N = total number of functions and n ((-) = number of functions having exactly t elements in the range. \[N={{3}^{n}},n(1)={{3.2}^{n}},n(2)=3,n(3)=0\] \[\therefore \]\[M=({{3}^{n}}-{{3.2}^{n}}+3+0)\] Hence, total number of favourable cases \[={{({{3}^{n}}-{{3.2}^{n}}+3)}^{6}}{{C}_{3}}\] Total number of cases\[={{6}^{n}}\] \[\therefore \] Required probability \[=\frac{{{({{3}^{n}}-{{3.2}^{n}}+3)}^{6}}{{C}_{3}}}{{{6}^{n}}}\]
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