Question

If three dice are thrown simultaneously, then the probability of getting a score of 7 is

• A.
• B.
• C.
• D. None of these

Answer 1

The correct option is C

$n\left(S\right)=6\times 6\times 6$
n(E) = The number of solutions of x + y + z = 7
where $1\le x\le 6,1\le y\le 6,1\le z\le 6$
= coefficient of $x^7$ in $(x+x^2+\dots \dots +x^6{)}^3$
= coefficient of $x^4$ in $(1+x+\dots \dots +x^5{)}^3$
= coefficient of $x^4$ in ${\left(\frac{1-x^6}{1-x}\right)}^3$
= coefficient of $x^4$ in $(1–3.x^6+3.x^{12}–x^{18})(1–x{)}^{-3}$
= coefficient of $x^4$ in $(1–3x^6+3x^{12}–x^{18})$
${(}^2{C}_0{+}^3{C}_1{+}^4{C}_2{x}^2{+}^5{C}_3{x}^3{+}^6{C}_4{x}^4+....)$
${=}^6{C}_4=\frac{10!}{4!.2!}=\frac{6\times 5}{2}=15$
$\therefore$ $p\left(E\right)=\frac{n\left(E\right)}{n\left(S\right)}=\frac{15}{6\times 6\times 6}=\frac{5}{72}$