A pair of dice is thrown 5 times. For each throw, a total of 5 is considered a success. If the probability of at least 4 successes is \frac{k}{3^{11}}, then k is equal to :

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Answer: C. 123 Given, a pair of dice is thrown once, then number of outcomes =6 	imes 6=36 A total of 5 are \{(1,4),(2,3),(4,1),(3,2)\} \begin{aligned} & 	herefore p =p(	ext { success })=\frac{4}{36}=\frac{1}{9} \\ & q =1-\frac{1}{9}=\frac{8}{9} \end{aligned} Let X represent the number of success \begin{aligned} & 	herefore P(	ext { at least } 4 	ext { success }) \\ &=P(X=4)+P(X=5) \\ &={ }^5 C_4 p^4 q+{ }^5 C_5 p^5 q^{5-5} \\ &=5\left(\frac{1}{9}ight)^4\left(\frac{8}{9}ight)+\left(\frac{1}{9}ight)^5 \\ &=\frac{40+1}{\left(3^2ight)^5}=\frac{41}{310} \\ &=\frac{41 	imes 3}{3^{11}}=\frac{123}{3^{11}} \\ & 	herefore k=123 \end{aligned}

A pair of dice is thrown 5 times. For each throw, a total of 5 is considered a success. If the probability of at least 4 successes is $$\frac{k}{3^{11}}$$, then $$k$$ is equal to :

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