Question

Assertion :If three dice are thrown simultaneously then probability of getting a sum of $15$ is $\frac{5}{108}$. Reason: The number of sample point of sample space is 216 and the number of favourable points are equal to the number of solutions $x+y+z=15$ where $1\le x,y,z\le 6$.

• A. Both Assertion & Reason are individually true & Reason is correct explanation of Assertion
• B. Both Assertion & Reason are individually true but Reason is not the ,correct (proper) explanation of Assertion
• C. Assertion is true but Reason is false
• D. Assertion is false but Reason is true

Answer 1

The correct option is A Both Assertion & Reason are individually true & Reason is correct explanation of Assertion

$E=$ Number of solutions of $x+y+z=15$
$=$ coefficient of $x^{15}in{(x+x^2+...x^6)}^3$
$=$ coefficient of $x^{12}in{(1+x+...x^5)}^3$
$=$ coefficient of $x^{12}in(1-3x^6+3x^{12}-x^{18}){(1-x)}^3$
$=$coefficient of $x^{12}in(1-3x^6+3x^{12}-x^{18})({}^2{\text{C}}_0{+}^3{\text{C}}_{1}x{+}^4{\text{C}}_2{x}^2...)$
${=}^{14}{\text{C}}_{12}-3{\times }^8{\text{C}}_6+3{\times }^2{\text{C}}_0$ $=10$
$\therefore P\left(E\right)=\frac{n\left(E\right)}{n\left(S\right)}=\frac{10}{216}=\frac{5}{108}$
Assertion (A) & Reason (R) both are correct and Reason
(R) is the correct (proper) explanation of Assertion (A).