Question

Three different dice are rolled simultaneously three times. The probability that all of them show different numbers only two times, is equal to

• A. $100/729$
• B. $2/3$
• C. $100/243$
• D. $175/324$

Answer 1

The correct option is D $175/324$

The dice will show different numbers twice if either in one throw they show the same number or any two of them show the same number and in two throws they all show different numbers.
In one throw they show the same number and in two throws they all show different numbers:
${}_1^{3}C\ast \left\{\right(\frac{1}{6})\ast (1)\ast (1\left)\right\}\ast {\left\{\right(1)\ast (\frac{5}{6})\ast (\frac{4}{6}\left)\right\}}^2=\frac{25}{162}$
In one throw any two of them show the same number and in two throws they all show different numbers:
${}_1^{3}C\ast \{{}_2^{3}C\ast (\frac{1}{6})\ast (1)\ast (\frac{5}{6}\left)\right\}\ast {\left\{\right(1)\ast (\frac{5}{6})\ast (\frac{4}{6}\left)\right\}}^2=\frac{125}{324}$

Hence, the required probability $=\frac{25}{162}+\frac{125}{324}=\frac{175}{324}$