Question

find the probability of exactly three successes in eight trials of a binomial experiment in which the probability of success is 45%.
p(3) = _8c_3(0.45)^3(0.55)^{8 - 3}
solve part of the answer.
_8c_3 = ?

Explanation:

Step1: Recall combination formula

The combination formula is \(_{n}C_{r}=\frac{n!}{r!(n - r)!}\). Here \(n = 8\) and \(r=3\).

Step2: Calculate factorial values

\(n!=8! = 8\times7\times6\times5\times4\times3\times2\times1=40320\), \(r!=3! = 3\times2\times1 = 6\), \((n - r)!=(8 - 3)!=5! = 5\times4\times3\times2\times1=120\).

Step3: Substitute into formula

\(_{8}C_{3}=\frac{8!}{3!(8 - 3)!}=\frac{8\times7\times6\times5!}{3!\times5!}\). Since \(5!\) cancels out, we have \(\frac{8\times7\times6}{3\times2\times1}\).
\(8\times7\times6 = 336\), \(3\times2\times1=6\). Then \(\frac{336}{6}=56\).

Answer:

\(56\)