Combinations TG pp 259 and LM pp. 301-317 DEPARTMENT OF EDUCATION Recall: PRIMING ACTIVITY How are you? Each group should familiarize every member by making handshake with each other. Count the number of handshakes made then complete the table: DEPARTMENT OF EDUCATION Answer How are you?( individual assessment) Number of persons ii Number of handshakes 2 ____ 3 ____ 4 ____ 5 ____ 6 ____ 7 ____ DEPARTMENT OF EDUCATION Answer How are you? Number of persons Number of handshakes 2 1 3 3 4 6 5 10 6 15 n ____ DEPARTMENT OF EDUCATION Answer How are you? Number of persons Number of handshakes 2 1 2!/(2-2)! 2! 3 3 3!/(3-2)! 2! 4 6 4!/ (4-2)! 2! 5 10 5!/ (5-2)! 2! 6 15 6!/ (6-2)! 2! n ____ n!/ (n-r)! r! DEPARTMENT OF EDUCATION Combinations- refers to the number of ways of selecting from a set n the order is not important. n! C(n,r) = (n - r)! r! The number of combinations of n objects taken r at a time denoted by π π πͺ π, π , ππͺπ , πΆπ ππ πΆ π DEPARTMENT OF EDUCATION Activity 1 1. In how many ways can a committee consisting of 4 members be formed from 8 people? 2. How many polygons can be possibly formed from 6 distinct points on a plane, no three of which are collinear? DEPARTMENT OF EDUCATION Activity 1 1. In how many ways can a committee consisting of 4 members be formed from 8 people? C= 8!/(8-4)!4!= 70 2. How many polygons can be possibly formed from 6 distinct points on a plane, no three of which are collinear? The polygon may have 3, 4, 5, or 6 vertices. Thus the number N possible polygons is: π = πΆ 6,3 + πΆ 6,4 + πΆ 6,5 + πΆ 6,6 6β5β4 6β5β4β3 6β5β4β3β2 6β5β4β3β2β1 = + + + 3! 4! 5! 6! =5β4+5β3+6+1 = 20 + 15 + 7 π = 42 possible polygons DEPARTMENT OF EDUCATION What to Reflect on and Understand? Activity 2: Reason Out 1. How do you determine if a situation or problem involves permutations or combinations? 2. How do you find the number of combinations of a. n objects taken r at a time b. n objects when taken all at a time? DEPARTMENT OF EDUCATION What to Reflect on and Understand? Activity 2: Reason Out 1. How do you determine if a situation or problem involves permutations or combinations? A situation or problem involves PERMUTATIONS if the order of the objects is important, if NOT then it is COMBINATIONS. 2. How do you find the number of combinations of a. n objects taken r at a time C(n,r) = n!/(n-r)!(r)! b. n objects when taken all at a time? C(n,n) = n!/(n-n)! (n)! = 1 DEPARTMENT OF EDUCATION Assessment Flex That Brain Solve the ff. problems completely: 1. If there are12 teams in a basketball tournament and each team must play every other team in the eliminations, how many elimination games will there be? 2. If there are 7 distinct points in a plane with no three of which are collinear, how many different polygons can be possibly formed? 3. In how many ways can a committee of 5 be formed from 5 juniors and 7 seniors if the committee must have 3 seniors? DEPARTMENT OF EDUCATION Assessment Flex That Brain Solve the ff. problems completely: 1. If there are12 teams in a basketball tournament and each team must play every other team in the eliminations, how many elimination games will there be? C= (12, 2) = 66 games 2. If there are 7 distinct points in a plane with no three of which are collinear, how many different polygons can be possibly formed? C (total) = C(7,3) + C(7,4) +C(7,5) + C(7,6)+ C(7,7) = 99 polygons 3. In how many ways can a committee of 5 be formed from 5 juniors and 7 seniors if the committee must have 3 seniors? C(total) = C(7,3) . C(5,2) = 350 committees DEPARTMENT OF EDUCATION Thank You DEPARTMENT OF EDUCATION