Question

Three six faced fair dice are thrown together. The probability that the sum of the numbers appearing on the dice is 8, is:

• A. $\frac{7}{72}$
• B. $\frac{7}{54}$
• C. $\frac{4}{25}$
• D. $None$

Answer 1

The correct option is A $\frac{7}{72}$

Number of ways in which we get the sum equal to 8 with 3 dice =
=coefficient of $x^8$ in $[x+x^2+......+x^6{]}^3$
=coefficient of $x^5$ in $[1+x+....+x^5{]}^3$
=coefficient of $x^5$ in $(1-x^6{)}^3(1-x{)}^{-3}$
=coefficient of $x^5$ in $(1-x{)}^{-3}$
$=\frac{\mathrm{3.4.5.6.7}}{\mathrm{1.2.3.4.5}}=21$
Also total number of ways of throwing 3 dice
$=6^3=216$
$\therefore$ Probability =$\frac{21}{216}=\frac{7}{72}$