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)=_{8}c_{3}(0.45)^{3}(0.55)^{8 - 3}
solve part of the answer.
_{8}c_{3}=\square{?}

Explanation:

Step1: Recall the combination formula

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

Step2: Calculate factorial values

\(n!=n\times(n - 1)\times\cdots\times1\). So \(8! = 8\times7\times6\times5\times4\times3\times2\times1=40320\), \(3! = 3\times2\times1 = 6\), and \((8 - 3)!=5! = 5\times4\times3\times2\times1=120\).

Step3: Substitute into the combination formula

\(_8C_3=\frac{8!}{3!(8 - 3)!}=\frac{40320}{6\times120}\).
\(\frac{40320}{6\times120}=\frac{40320}{720}=56\).

Answer:

\(56\)