The Fundamental Counting Principal, Permutations, & Combinations

The Fundamental Counting Principal If you have 2 events: 1 event can occur m ways and another event can occur n ways, then the number of ways that both can occur is m*n Event 1 = 4 types of meats Event 2 = 3 types of bread How many diff types of sandwiches can you make? 4*3 = 12

3 or more events: 3 events can occur m, n, & p ways, then the number of ways all three can occur is m*n*p 4 meats 3 cheeses 3 breads How many different sandwiches can you make? 4*3*3 = 36 sandwiches

At a restaurant at Cedar Point, you have the choice of 8 different entrees, 2 different salads, 12 different drinks, & 6 different deserts. How many different dinners (one choice of each) can you choose? 8*2*12*6= 1152 different dinners

Fund. Counting Principal with repetition Ohio Licenses plates have 3 #’s followed by 3 letters. 1. How many different licenses plates are possible if digits and letters can be repeated? There are 10 choices for digits and 26 choices for letters. 10*10*10*26*26*26= 17,576,000 different plates

How many plates are possible if digits and numbers cannot be repeated? There are still 10 choices for the 1 st digit but only 9 choices for the 2 nd, and 8 for the 3 rd. For the letters, there are 26 for the first, but only 25 for the 2 nd and 24 for the 3 rd. 10*9*8*26*25*24= 11,232,000 plates

Phone numbers How many different 7 digit phone numbers are possible if the 1 st digit cannot be a 0 or 1? 8*10*10*10*10*10*10= 8,000,000 different numbers

Testing A multiple choice test has 10 questions with 4 answers each. How many ways can you complete the test? 4*4*4*4*4*4*4*4*4*4 = 4 10 = 1,048,576

Using Permutations An ordering of n objects is a permutation of the objects.

There are 6 permutations of the letters A, B, &C ABC ACB BAC BCA CAB CBA You can use the Fund. Counting Principal to determine the number of permutations of n objects. Like this ABC. There are 3 choices for 1 st # 2 choices for 2 nd # 1 choice for 3 rd. 3*2*1 = 6 ways to arrange the letters

In general, the # of permutations of n objects is: n! = n*(n-1)*(n-2)* …

12 skiers… How many different ways can 12 skiers in the Olympic finals finish the competition? (if there are no ties) 12! = 12*11*10*9*8*7*6*5*4*3*2*1 = 479,001,600 different ways

Factorial with a calculator: Hit math then over, over, over. Option 4

Back to the finals in the Olympic skiing competition. How many different ways can 3 of the skiers finish 1 st, 2 nd, & 3 rd (gold, silver, bronze) Any of the 12 skiers can finish 1 st, the any of the remaining 11 can finish 2 nd, and any of the remaining 10 can finish 3 rd. So the number of ways the skiers can win the medals is 12*11*10 = 1320

Permutation of n objects taken r at a time n P r =

Back to the last problem with the skiers It can be set up as the number of permutations of 12 objects taken 3 at a time. 12 P 3 = 12! = 12! = (12-3)!9! 12*11*10*9*8*7*6*5*4*3*2*1 = 9*8*7*6*5*4*3*2*1 12*11*10 = 1320

10 colleges, you want to visit all or some. How many ways can you visit 6 of them: Permutation of 10 objects taken 6 at a time: 10 P 6 = 10!/(10-6)! = 10!/4! = 3,628,800/24 = 151,200

How many ways can you visit all 10 of them: 10 P 10 = 10!/(10-10)! = 10!/0!= 10! = ( 0! By definition = 1) 3,628,800

So far in our problems, we have used distinct objects. If some of the objects are repeated, then some of the permutations are not distinguishable. There are 6 ways to order the letters M,O,M MOM, OMM, MMO Only 3 are distinguishable. 3!/2! = 6/2 = 3

Permutations with Repetition The number of DISTINGUISHABLE permutations of n objects where one object is repeated q 1 times, another is repeated q 2 times, and so on : n! q 1 ! * q 2 ! * … * q k !

Find the number of distinguishable permutations of the letters: OHIO : 4 letters with 0 repeated 2 times 4! = 24 = 12 2! 2 MISSISSIPPI : 11 letters with I repeated 4 times, S repeated 4 times, P repeated 2 times 11! = 39,916,800 = 34,650 4!*4!*2!24*24*2

Find the number of distinguishable permutations of the letters: SUMMER : 360 WATERFALL : 90,720

A dog has 8 puppies, 3 male and 5 female. How many birth orders are possible 8!/(3!*5!) = 56

What are Combinations?

Combinations and Permutations Whats the Difference? In English we use the word "combination" loosely, without thinking if the order of things is important. In other words:

Whats the Difference? "My fruit salad is a combination of apples, grapes and bananas" We don't care what order the fruits are in, they could also be "bananas, grapes and apples" or "grapes, apples and bananas", its the same fruit salad. "The combination to the safe was 472". Now we do care about the order. "724" would not work, nor would "247". It has to be exactly

So, in Mathematics we use more precise language: If the order doesn't matter, it is a Combination. If the order does matter it is a Permutation. So, we should really call this a "Permutation Lock"!

How lotteries work. The numbers are drawn one at a time, and if you have the lucky numbers (no matter what order) you win! So what is your chance of winning?

The easiest way to explain it is to: assume that the order does matter (ie permutations), then alter it so the order does not matter.

Wow, that was pretty cool! But knowing how these work is only half of it. Figuring out how to interpret a real world situation can be a challenging. But at least now you know how to find all variations of combinations, permutations, and the counting principle.