Question

four girls and six boys are in a spanish club. three of the people will be chosen at random to represent the group in a photograph.
what is the probability that one girl and two boys will be chosen?
40%
50%
60%
70%

Explanation:

Step1: Calculate total number of ways to choose 3 people

Use combination formula \(C(n,r)=\frac{n!}{r!(n - r)!}\), where \(n=4 + 6=10\), \(r = 3\).
\(C(10,3)=\frac{10!}{3!(10-3)!}=\frac{10\times9\times8}{3\times2\times1}=120\)

Step2: Calculate number of ways to choose 1 girl and 2 boys

Number of ways to choose 1 girl: \(C(4,1)=\frac{4!}{1!(4 - 1)!}=4\)
Number of ways to choose 2 boys: \(C(6,2)=\frac{6!}{2!(6-2)!}=\frac{6\times5}{2\times1}=15\)
By multiplication principle, number of ways to choose 1 girl and 2 boys is \(C(4,1)\times C(6,2)=4\times15 = 60\)

Step3: Calculate probability

Probability \(P=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}=\frac{60}{120}=0.5\)
Convert to percentage: \(0.5\times100\% = 50\%\)

Answer:

50%