Question

Ten items out of a set of $100$ are defective. What is the probability that $3$ out of any four chosen are defective?

Answer 1

Probability$=\frac{{}^{90}{C}_1{}^{10}{C}_3}{{}^{100}{C}_4}$

$=\frac{\frac{90!}{(90-1)!1!}\times \frac{10!}{(10-3)!3!}}{\frac{100!}{(100-4)!4!}}$

$=\frac{\frac{90!}{89!1!}\times \frac{10!}{7!3!}}{\frac{100!}{96!4!}}$

$=\frac{\frac{90\times 89!}{89!}\times \frac{10\times 9\times 8\times 7!}{7!3!}}{\frac{100\times 99\times 98\times 97\times 96!}{96!\times 4\times 3\times 2}}$

$=\frac{90\times 10\times 3\times 4}{25\times 33\times 49\times 97}$

$=\frac{6\times 2\times 3\times 4}{11\times 49\times 97}$

$=\frac{144}{52283}$