Teams

471 /www/nrich/html/content/03/05/15plus2/ 1 2011-02-01T00:00:01 <mdoxml version="1.0"> <mdo:image align="right" alt="Chess board image" height="160" src="chess_m_f.gif" width="160"></mdo:image> A chess club has $10$ members, $4$ women and $6$ men, of whom two are brothers. A team of $4$ members is selected to play a match. Suppose that each member has an equal chance of being selected what is the probability that both brothers are in the team and what is the probability that neither brother is in the team? What is the probability that there are at least two women in the team and what is the probability that there are at least two men in the team? </mdoxml> <mdoxml version="1.0"> Phil sent us this solution: There are $^{10}C_4=210$ possible chess teams. <ol> <li>To find the number of teams containing both brothers, we need the number of ways of choosing the other $2$ team members from the remaining $8$ people, which is $^8C_2=28$. So the probability is $28/210=2/15$.</li> <li>To find the number of teams containing neither brother, we need the number of ways of choosing all $4$ team members from the remaining $8$ people, which is $^8C_4=70$. So the probability is $70/210=1/3$.</li> <li>To find the number of teams containing at least $2$ women, we can take $210$ - the number of teams containing $0$ women - the number of teams containing $1$ woman. There are $^6C_4=15$ teams with $0$ women, and $4 \times$ $^6C_3=80$ with $1$ woman, so the required probability is $1-\frac{15}{210}-\frac{80}{210}=\frac{ 115}{210}=\frac{23}{42}$.</li> <li>Similar to the last one. There is only $1$ team containing $0$ men. There are $4\times$ $^6C_1=24$ teams containing $1$ man, so the probability we want is $1-\frac{1}{210}-\frac{24}{210}=\frac{185}{210}=\frac{37}{42}$.</li> </ol> </mdoxml> <mdoxml version="1.0"> How many ways can you choose a team of $4$ from $10$? </mdoxml> 2 3 0 0 0 0 1 Teams Two brothers belong to a club with 10 members. Four are selected for a match. Find the probability that both brothers are selected. Probability Theoretical probability Probability Equally likely outcomes Decision Mathematics and Combinatorics Combinations