a Since the selection order is not important, each our selection is a combination.
B
b Since the selection order is not important, each our selection is a combination.
C
c Find the ratio between the number of boards that have Mariko as a member and the total number of possible boards.
A
a _(900) C_(12) ≈ 4.48 * 10^(26)
B
b _(899) C_(11) ≈ 7.30 * 10^(26)
C
c ≈ 1.63 %
Practice makes perfect
a We know that 12 out of 900 students are chosen for the Judicial Board for minor student infractions. When we find the number of different Judicial Boards, the order in which the members were selected is not important. This means each our selection is a combination.
The number of combinations _n C_r is the number of ways we can choose r elements from n elements when the selection order is not important.
In our case, n= 900, since we choose the board from 900 students, and r= 12, since any chosen board contains 12 students. This means we have _(900) C_(12) ways of choosing a board. We can put this expression into a calculator to get its approximate value.
_(900) C_(12) ≈ 4.48 * 10^(26)
b We know that Mariko wants to be chosen as a member of the board. Since the order in which the members of the board are chosen is not important, each such selection is a combination.
The number of combinations _n C_r is the number of ways we can choose r elements from n elements when the selection order is not important.
Let's say Mariko was chosen as a member. The board needs 11 other members, so r = 11. Since Mariko is a student, we choose 11 students from all students except Mariko. As a result, n = 1000-1 = 900. This means there are _(899) C_(11) ways of choosing the board that have Mariko as a member. Let's approximate this number.
_(899) C_(11) ≈ 7.30 * 10^(24)
c We want to find the probability that Mariko will be chosen as a member of a board. To do so, we can divide the number of boards that have Mariko as a member by the total number of boards.
P(choosing Mariko) = No. of boards with Mariko/No. of boardsIn Part A, we found that there are _(900) C_(12), or about 4.48 * 10^(26), possible boards. From Part B we have that there are _(899) C_(11), or about 7.30 * 10^(24), possible boards that have Mariko as a member. Let's substitute these values into our formula for the probability.
P(choosing Mariko) = _(899) C_(11)/_(900) C_(12)
Let's express this probability as a percent. To do so, we can use the approximations from previous Parts.
P(choosing Mariko) = No. of boards with Mariko/No. of boards
No. of boards with Mariko ≈ 7.30 * 10^(24), No. of boards ≈ 4.48 * 10^(26)
