P Permutations versus Combinations
 4! is read as 4 factorialand is...4 x 3 x 2 x 1.  It equals24.3! = 5! is read as 5 factorialand is...5 x 4 x 3 x 2 x 1.  It equals 120.7! = 7 x 6 x 5 x     ••• 3! =         x        x  0! is read as 0 factorialand equals 1. 3! is read as 3 factorialand is...3 x 2 x 1.  It equals 6.4! =       x      x     x     Permutations - order matters like combination locks Of n things taken r at a time.nPr=           orn count down r times(n-r)!n!     N is the number of items being arranged in lots of r.  How many items are there to pick from           ?nPr=           orn count down r times(n-r)!n!     N is the number of items being arranged in lots of r.  How many items are in each group?         itemsnPr=           orn count down r times(n-r)!n!   For this problem start at 6 and count down two place.Permutations - order matters like combination locks  -  Of n things taken r at a time.nPr= 6 x 5n multiplied by count down r places(n-r)!n!or6P2= (6-2)!6!   Permutations - order matters like combination locks  -  Of n things taken r at a time.nPr6 x 5 x 4 ==                  orn count down r times(n-r)!n!6P3= (6-3)!6!   Permutations - order matters like combination locks  -  Of n things taken r at a time.nPr7 x 6 x 5 ==                 orn count down r times(n-r)!n!7P3= (7-3)!7!   Permutations - order matters like combination locks  -  Of n things taken r at a time.5P2= or 5 x 4 = 20(5-2)!5!= 5!!    Permutations - order matters like combination locks  -  Of n things taken r at a time.8P2= count down 2 spots8 x 7 = (8-2)!8!= 8!!    Permutations - order matters like combination locks  -  Of n things taken r at a time.8P3   x     x      =  336= count down 3 spots(8-3)!8!= 8!!    Permutations - order matters like combination locks  -  Of n things taken r at a time.8P4   x     x     x    =  1680= count down 4 spots(8-4)!8!= 8!!       Combinations - order does not matter - like orderinga pizza with multiple toppings - of n things taken r at a time.nn count down r timesCr=r!(n-r)!n!r!or       Combinations - order does not matter - like orderinga pizza with multiple toppings - of n things taken r at a time.nn count down r times   would be like the number of toppings to pick fromCr=r!(n-r)!n!r!or       Combinations - order does not matter - like orderinga pizza with multiple toppings - of n things taken r at a time.nn count down r timesC  is like the number of toppings you can selectr=r!(n-r)!n!r!or             Combinations - order does not matter - like orderinga pizza with multiple toppings - of n things taken r at a time.orn count down r times6C2=r!2!(6-2)!6!=(2•1)6•5•4•3•2•136 x 52 x 1(4•3•2•1)=         Combinations - order does not matter - like orderinga pizza with multiple toppings - of n things taken r at a time.orn count down r times6C3=r!3!(6-3)!6!=6 x 5 x 43 x 2 x 1(3•2•1)6•5•4•3•2•1(3•2•1)=         Combinations - order does not matter - like orderinga pizza with multiple toppings - of n things taken r at a time.orn count down r times7C3=r!3!(7-3)!7!=(3•2•1)7•6•5•4•3•2•17 x 6 x 53 x 2 x 1(4•3•2•1)=      Combinations - order does not matter - like orderinga pizza with multiple toppings - of n things taken r at a time.5C2=2!(5-2)!5!orn count down r timesr!      Combinations - order does not matter - like orderinga pizza with multiple toppings - of n things taken r at a time.8C2=2!(8-2)!8!orn count down r timesr!      Combinations - order does not matter - like orderinga pizza with multiple toppings - of n things taken r at a time.5C3=3!(5-3)!5!orn count down r timesr! Where a permutation of 3 things taken in lots of 3 has 6 choices (see choices below) because order matters,a combination of 3 things taken in lots of 3 has         choice(s) because order does not matter.ABC     ACBBAC     BCACAB     CBAPossible choices for a   P   33 Where a permutation of 3 things taken in lots of 2 has 6 choices (see choices below) because order matters,a combination of 3 things taken in lots of 2 has         choice(s) because order does not matter.AB         ACBA         BCCA         CBPossible choices for a   P   32 Where a combination of 4 things taken in lots of 2 has 6 choice(s) because order does not matter a permutation will have more choices as when the orderchanges it makes a new set.  The permutation of 4things taken in lots of 2 will have          choices.1) AB = BA2) AC = CA3) AD = DAPossible choices for a   C   4) BC = CB5) BD = DB6) CD = CD42 Runners finishing a raceOrder matters as in (AB ≠ BA)  PermutationOrder doesn't matter as in (AB = BA) Combination Four people applying for the same two jobsOrder matters as in (AB ≠ BA)  PermutationOrder doesn't matter as in (AB = BA) Combination Four people applying for two different jobsOrder matters as in (AB ≠ BA)  PermutationOrder doesn't matter as in (AB = BA) Combination Your locker combinationOrder matters as in (AB ≠ BA)  PermutationOrder doesn't matter as in (AB = BA) Combination Ordering pizza with several toppingsOrder matters as in (AB ≠ BA)  PermutationOrder doesn't matter as in (AB = BA) Combination Applying for the same jobs on a year book staffOrder matters as in (AB ≠ BA)  PermutationOrder doesn't matter as in (AB = BA) Combination Four people applying for different positions on theyearbook staffOrder matters as in (AB ≠ BA)  PermutationOrder doesn't matter as in (AB = BA) Combination Running for office - first place votes is President, second place, Vice President, etc.Order matters as in (AB ≠ BA)  PermutationOrder doesn't matter as in (AB = BA) Combination
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