Question

Five dice are rolled at random and the sum of the numbers shown on them is $16$. Find the probability that the numbers shown on each is any one from $2$ ,$3$,$4$ or $5$?

• A. $\frac{7}{49}$
• B. $\frac{7}{48}$
• C. $\frac{9}{49}$
• D. $\frac{9}{48}$

Answer 1

The correct option is C $\frac{9}{49}$

Given that $5$ dice are rolled randomly.

So the possible outcomes are $6^5$ i.e, $7776$ outcomes.

Sum the possible outcomes achived in the following way :

The no. shown on each should be any one from $2,3,4,$ or $5.$

So, the sum of $16$ can be obtained in,

$1.)5,5,2,2,2=\frac{5!}{2!3!}=10$

$2.)2,2,3,4,5=\frac{5!}{2!}=60$

$3).3,3,3,3,4=\frac{5!}{4!}=5$

$4).4,4,4,2,2=\frac{5!}{3!2!}=10$

$5).3,3,3,5,2=\frac{5!}{3!}=20$

$\Rightarrow$ Total $=10+60+5+10+20=105$

$\therefore$ Required probability $=\frac{105\times (2+3+4+5)}{6^5}$

$=\mathrm{0.1836.}$

Hence, the answer is $\frac{9}{49}.$