A) \[\frac{5}{216}\]
B) \[\frac{1}{6}\]
C) \[\frac{5}{72}\]
D) None of these
Correct Answer: C
Solution :
\[n(S)=6\times 6\times 6\] \[n(E)=\] The number of solutions of \[x+y+z=7,\] where \[1\le x\le 6,\] \[1\le y\le 6,\] \[1\le z\le 6\] = coefficient of \[{{x}^{7}}\] in \[{{(x+{{x}^{2}}+.......+{{x}^{6}})}^{3}}\] = coefficient of \[{{x}^{4}}\] in \[{{(1+x+......+{{x}^{5}})}^{3}}\] = coefficient of \[{{x}^{4}}\]in \[{{\left( \frac{1-{{x}^{6}}}{1-x} \right)}^{3}}\] = coefficient of \[{{x}^{4}}\] in \[(1-3.{{x}^{6}}+3.{{x}^{12}}-{{x}^{18}}){{(1-x)}^{-3}}\] = coefficient of \[{{x}^{4}}\] in \[(1-3{{x}^{6}}+3{{x}^{12}}-{{x}^{18}})\] \[({}^{2}{{C}_{0}}+{}^{3}{{C}_{1}}x+{}^{4}{{C}_{2}}{{x}^{2}}+{}^{5}{{C}_{3}}{{x}^{3}}+{}^{6}{{C}_{4}}{{x}^{4}}+.......)\] = \[{}^{6}{{C}_{4}}=\frac{6\,\,!}{4\,\,!\,.\,2\,\,!}=\frac{6\times 5}{2}=15\] \ \[p(E)=\frac{n(E)}{n(S)}=\frac{15}{6\times 6\times 6}=\frac{5}{72}.\]
You need to login to perform this action.
You will be redirected in
3 sec
