Permutation and Combination
Permutation : Permutation means arrangement of things. The word arrangement is used, if the order of things is considered.
Combination: Combination means selection of things. The word selection is used, when the order of things has no importance.
Example: Suppose we have to form a number of consisting of three digits using the digits 1,2,3,4, To form this number the digits have to be arranged. Different numbers will get formed depending upon the order in which we arrange the digits. This is an example of Permutation.
Now suppose that we have to make a team of 11 players out of 20 players, This is an example of combination, because the order of players in the team will not result in a change in the team. No matter in which order we list out the players the team will remain the same! For a different team to be formed at least one player will have to be changed.
before start topic understand Factorial
Let n be a positive integer. Then, factorial n, denoted n! is defined as:
n! = n(n – 1)(n – 2) … 3.2.1.
- We define 0! = 1.
- 4! = (4 x 3 x 2 x 1) = 24.
- 5! = (5 x 4 x 3 x 2 x 1) = 120.
The different arrangements of a given number of things by taking some or all at a time, are called permutations.
- All permutations (or arrangements) made with the letters a, b, c by taking two at a time are (ab, ba, ac, ca, bc, cb).
- All permutations made with the letters a, b, c taking all at a time are:
( abc, acb, bac, bca, cab, cba)
Number of all permutations of n things, taken r at a time, is given by:
nPr = n(n – 1)(n – 2) … (n – r + 1) =n! / (n – r)!
- 6P2 = (6 x 5) = 30.
- 7P3 = (7 x 6 x 5) = 210.
- Cor. number of all permutations of n things, taken all at a time = n!.
An Important Result:
If there are n subjects of which p1 are alike of one kind; p2 are alike of another kind; p3 are alike of third kind and so on and pr are alike of rth kind,
such that (p1 + p2 + … pr) = n.
Then, number of permutations of these n objects is = n! / (p1!).(p2)!…..(pr!)
Each of the different groups or selections which can be formed by taking some or all of a number of objects is called a combination.
- Suppose we want to select two out of three boys A, B, C. Then, possible selections are AB, BC and CA.Note: AB and BA represent the same selection.
- All the combinations formed by a, b, c taking ab, bc, ca.
- The only combination that can be formed of three letters a, b, c taken all at a time is abc.
- Various groups of 2 out of four persons A, B, C, D are:
AB, AC, AD, BC, BD, CD.
- Note that ab ba are two different permutations but they represent the same combination.
Number of Combinations:
The number of all combinations of n things, taken r at a time is:
nCn = n! / (r!)(n-r)!
nCn = 1 and nC0 = 1.
nCr = nC(n – r)
- 11C4=(11 x 10 x 9 x 8) / (4 x 3 x 2 x 1) = 330
Problem Practice: click here