a dice is thrown 5 times , then the probability that an even number will come up exactly 3 times is

Dear Student,

As the dice is thrown 5 times and each throw can result in 6 outcomes.
The sample space will be n(S)=6^5.

Now let n(A) be the event of getting an even number in a throw of dice, so 3 out of 6 possible outcome. 
we need 3 number of favourable outcomes in 5 throws=3³ 

But the even number can occur in any three of the 5 throws of dice. Hence we need to select 3 out of 5 throws of dice

$$\begin{gathered} n\left(A\right)={}^{5}C_3\times 3^3 \\ =\frac{5\times 4}{2}\times 3^3 \\ =10\times 27 \\ 270 \end{gathered}$$


Probability p(A)=$\frac{n\left(A\right)}{n\left(S\right)}$
$$\begin{gathered} \frac{270}{6^5} \\ =\frac{270}{216\times 6\times 6} \\ =\frac{45}{6\times 216} \\ =\frac{5}{6\times 24} \\ =\frac{5}{144} \end{gathered}$$

Regards,