Combinations Math 10 PDF

COMBINATIONS MATH 10  How many permutations can you make with the red ball and the yellow ball? 2  If order is not important, how many arrangements can you make with the red ball and the yellow ball? 1  Different!...

COMBINATIONS MATH 10  How many permutations can you make with the red ball and the yellow ball? 2  If order is not important, how many arrangements can you make with the red ball and the yellow ball? 1  Different! Same! Permutation Combination red, yellow yellow, red A combination is a way of selecting r objects out of n objects where arrangement is not important. For each case, tell whether arrangement is important or not. Case 1 A group of 4 students needs to select a leader and an assistant leader from among them. How many arrangements of group leaders are possible? Case 2 A group of 4 students needs to select two representatives from among them. How many arrangements of representatives are possible? Suppose you have a red ball, a yellow ball, and a green ball. If you are going to choose two balls, how many combinations can you have? first second third combination combination combination The combination of n objects taken r at a time is denoted by 𝒏𝑪𝒓. 𝒏𝑷𝒓 = 𝒏𝑪𝒓 ∙ 𝒓! Solving for 𝒏𝑪𝒓 we get: 𝑷𝒓 𝒏! 𝒏𝑪𝒓 = = 𝒏 𝒓! 𝒏−𝒓 ! 𝒓! Hence, the number of combinations of three balls taken two at a time is given by 𝒏! 𝒏𝑪𝒓 = 𝒏−𝒓 !𝒓! 𝟑! 3𝑪𝟐 = 𝟑 − 𝟐 ! 𝟐! =𝟑 Example 1 From a class of 18 students, 8 students will be chosen to join the Overseas Learning Experience. In how many ways can the students be chosen? Solution 𝒏! 𝒏𝑪𝒓 = 𝒏 − 𝒓 ! 𝒓! 𝟏𝟖! 𝟏𝟖 ∙ 𝟏𝟕 ∙ 𝟏𝟔 ∙ 𝟏𝟓 ∙ 𝟏𝟒 ∙ 𝟏𝟑 ∙ 𝟏𝟐 ∙ 𝟏𝟏 ∙ 𝟏𝟎! 𝟏𝟖𝑪𝟖 = = = 𝟒𝟑 𝟕𝟓𝟖 𝟏𝟖 − 𝟖 ! 𝟖! 𝟏𝟎! 𝟖! Example 2 In how many ways can a team consisting of 2 boys and 3 girls be formed if 6 boys and 10 girls are qualified to be in the team? Solution 6! 10! 6∙5∙4! 10∙9∙8∙7! 6𝐶2 ∙ 10𝐶3 = ∙ 4!2! 7!3! = 4! 2! ∙ 7! 3! = 15 ∙ 120 = 1 800 ways Example 3 A committee of 3 members is to be formed from 6 women and 5 men. The committee must include at least 2 women. In how many ways can this be done? Solution or 𝟔𝑪𝟐 ∙ 𝟓𝑪𝟏 + 𝟔𝑪𝟑 ∙ 𝟓𝑪𝟎 6! 5! 6! 5! 6𝐶2 ∙ 5𝐶1 + 6𝐶3 ∙ 5𝐶0 = ∙ 4! 2! 1! 4! + ∙ 3! 3! 0! 5! = 15 ∙ 5 + 20 ∙ 1 = 75 + 20 = 95 ways Group Work Solve each problem. 1. For the official lineup of the GILAS basketball team, the coaching staff will select 12 out of 16 players. In how many ways can this be done? 2. In how many ways can a committee of 3 teachers and 3 students be formed if there are 5 teachers and 6 students to choose from? 3. A team of 5 is to be formed from 5 girls and 8 boys. If the team must include at least 3 boys, in how many ways can this be done? Group Work Solve each problem. 1. For the official lineup of the GILAS basketball team, the coaching staff will select 12 out of 16 players. In how many ways can this be done? Group Work Solve each problem. 2. In how many ways can a committee of 3 teachers and 3 students be formed if there are 5 teachers and 6 students to choose from? Group Work Solve each problem. 3. A team of 5 is to be formed from 5 girls and 8 boys. If the team must include at least 3 boys, in how many ways can this be done? Review Time 1. What is the difference between a permutation and a combination? 2. How do you find the combination of n objects taken r at a time? SEATWORK (20 points) p. 285, number 3, a – e (answers only) pp. 285 – 286, numbers 6 to 10 (formula-substitution-answer)