Permutation and Combination

TOPIC

PERMUTATION The different arrangement which can be made by taking some or all of a number of thing are called permutation. Suppose we have three objects a,b,c If we take two of these at time, then the arrangement are ab bc ca ba cb ac Thus the number of arrangement of 3 objects taken two at a time is 6. Each of these is called a permutation.

NOTATION OF PERMUTATION • If n and r are positive integers such that 1 < r < n, then the number all permutation of n distinct objects, taken r at a time is denoted by the symbol nPr or P(n,r). Thus, n

Pr or P(n,r)= Total number of permutation of n distinct objects, taken r at a time.

Examples • 6 permutation on a set of 3 letter taken 2 at a time i.e., 3

•

P2 = 6

permutation on a set of 4 objects taken 3 at a time i.e. 24

4

p3

= 24

Factorial Notation Recall the problem of counting how many ways we can seat three men in three chairs Because the product n x (n – 1) x … x 2 x 1 occurs often, we often write it in shorthand notation as n! The exclamation point is pronounced factorial n! means the product of n down to 1 3! = 3 · 2! = 3 · 2 · 1! = 3 · 2 · 1 = 6 1! AND 0! are both equivalent to 1 n! = n · (n – 1)! • We can expand a factorial into a product in order to quickly evaluate expressions containing factorials

FUNDAMENTAL PRINCIPLE OF COUNTING •

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 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

PERMUTATION OF ALIKE THINGS • The number of permutation of n things taken all at a time where p of the things are alike and of one kind, q there are alike and of another kind, r others are alike and of another kind and so on, is n! p! q! r!

CIRCULAR PERMUTATION

• CLOCK WISE PERMUTATION • ANTI CLOCK WISE PERMUTATION

COMBINATION The different group or selection which can be made by taking some or all of a number of things are called combination. For examples The combination which can be made by taking the letter a,b,c at a time are ab, bc, ac Thus in combination we are only concerned with selection group of things irrespective of the order of the things. The number of all combination of n things taken r at a time is denoted by nCr or c(n,r) or (nr) clearly nCr is denoted only when n and r are non-negative integers such that 0 < r < n.

EXAMPLE • How many five-cards hands are possible from a standard deck of cards? Sol. 52C5 = 2598960 • Suppose there are 15 girls an 18 boys in a class. In how many ways can 2 girls and 2 boys be selected for a group project? Sol. 15C2 X 18C2 = 16065