Question

a math teacher chooses 6 students at random from a class of 24 to solve a problem at the board. if 3 students in the entire class know how to solve the problem, what is the probability that all 6 students who were picked do not know how to solve the problem? note: for any final answer that has up to four decimal places, enter your answer without rounding the number. for any answers with more than four decimal values, round your final answer to four decimal places.

Explanation:

Step1: Calculate the number of students who don't know the solution

The total number of students is \(n = 24\), and the number of students who know the solution is \(k=3\). So the number of students who don't know the solution is \(24 - 3=21\).

Step2: Use the combination formula

The formula for combinations is \(C(n,r)=\frac{n!}{r!(n - r)!}\), where \(n\) is the total number of items, and \(r\) is the number of items to be chosen.
The number of ways to choose \(6\) students from \(24\) is \(C(24,6)=\frac{24!}{6!(24 - 6)!}=\frac{24!}{6!×18!}=\frac{24\times23\times22\times21\times20\times19}{6\times5\times4\times3\times2\times1}=134596\).
The number of ways to choose \(6\) students from the \(21\) students who don't know the solution is \(C(21,6)=\frac{21!}{6!(21 - 6)!}=\frac{21!}{6!×15!}=\frac{21\times20\times19\times18\times17\times16}{6\times5\times4\times3\times2\times1}=54264\).

Step3: Calculate the probability

The probability \(P\) that all \(6\) students don't know the solution is \(P=\frac{C(21,6)}{C(24,6)}\).
Substitute the values of \(C(21,6)\) and \(C(24,6)\) into the formula: \(P=\frac{54264}{134596}\approx0.4031\).

Answer:

\(0.4031\)