Total number of cubes=100

Number of cubes passed the test successfully=85

Number of failed in the test=15

It is given that if 10 cubes are selected at random to be inspected by the

company then we have to find the probability that 8 cubes will pass the test

and 2 cubes will fail in the test.

Total possible cases \(\displaystyle={100}{C}_{{{10}}}\)

Favorable cases \(\displaystyle=^{{85}}{C}_{{8}}\times^{{15}}{C}_{{2}}\)

Required probability \(\displaystyle={\frac{{{F}{a}{v}{{or}}{a}{b}le\ {c}{a}{s}{e}{s}}}{{{T}{o}{t}{a}{l}\ {p}{o}{s}{s}{i}{b}le\ {c}{a}{s}{e}{s}}}}\)

\(\displaystyle={\frac{{^{\left\lbrace{85}\right\rbrace}{C}_{{8}}\times^{{15}}{C}_{{2}}}}{{^{\left\lbrace{100}\right\rbrace}{C}_{{{10}}}}}}\)

\(\displaystyle{\frac{{{4.8125}\times{10}^{{{10}}}\times{105}}}{{{1.731}\times{10}^{{{13}}}}}}\)

\(\displaystyle={\frac{{{2.7801}\times{105}}}{{{1000}}}}\)

\(\displaystyle={\frac{{{291.92}}}{{{1000}}}}\)

\(\displaystyle={0.2919}\)

Thus, the required probability that 8 cubes will pass the test and 2 cubes will

\(\displaystyle{f}{a}{i}{l}\ in\ {t}{h}{e\ }{t}{e}{s}{t}\ {i}{s}\ {0.2919}.\)

Hence,option(e) is correct.