Basic Concept of Permutation and Combinations:
The Fundamental principle of counting:
If an event can occur in m different ways following which another event can occur in n different ways then the total occurrence of the events together can occur in m * n ways.
Permutations:
Suppose you want to arrange your paintings on a wall. There is only 1 way of arranging it if you have only one painting. Now suppose you have two paintings, P1 and P2 they can be arranged in 2 ways. Painting P1 first and P2 next, or P2 first and P1 next. Thus, there are two arrangements of the two books.
Let us assume that your younger brother also wants to add a painting P3 to the wall. After arranging P1 and P2 in one of the two ways, say P1P2, third painting can be arranged in following possible ways: P3P2P1, P2P3P1 or P2P1P3. Similarly, corresponding to P2P1, you have three other ways. So, the three paintings P1,P2 and P3 can be arranged in 3 * 2 = 6 ways.
The above example is an example of Permutation. Arrangement or Rearrangement of r things that can be done out of total n things is defined as Permutation?. It is denoted by nPr = n! / (n-r)!, where 0r n and the things do not repeat.
Note: The notation n! represents the product of first n natural numbers i.e the product 1*2*3*4………*(n-1)*n is denoted as n! . This symbol is read as n factorial.
Example1?: ?If you have 6 chocolates and you want to distribute them among 4 of your friends, in how many ways can this be done?
Solution:
Here we need to find a number of permutations of 4 objects out of 6 objects. Solution is ^{6}P_{4} i.e. ^{6}P_{4} = 6!/(6-4)! = 6(6−1)(6−2)(6−3)*2*1 / 2* 1= 6×5×4×3 = 360 Therefore, chocolates can be distributed in 360 ways.
Permutations under Some Conditions:
- The number of permutations of n different things, taken r at a time, when a particular thing is to be always included in each arrangement is: r^{n−1}P_{r−1} .
- The number of permutations of n different things, taken r at a time, when a particular thing is never taken in each arrangement is: ^{n−1}P_{r}
- The number of permutations of x different things, taken all at a time, when y specified things always come together is x!×(x−y+1)!
- The number of permutations of x different things, taken all at a time, when y specified never come together is: x!−[y!×(x−y+1)!]
- The number of permutations of n dissimilar things taken r at a time when k(< r) particular things always occur is: [^{n−k}P_{r−k} ]×[^{r}P_{k}]
- The number of permutations of n dissimilar things taken r at a time when k particular things never occur is:^{ n−k}P_{r}
- The number of permutations of n dissimilar things taken r at a time when repetition of things is allowed any number of times is: n^{r }
Combinations : Let us take an example of shirts and trousers you have 4 sets of bat and ball and you want to take 2 sets with you while going for a tournament. In how many ways can you do it?
The sets are Set1,Set2,Set3,Set4 Then you can choose 2 sets in the following ways:
{Set1,Set2} {Set1,Set3} {Set1,Set4}
{Set2,Set3} {Set2,Set4} {Set3,Set4}
Here {Set1,Set2} is the same as {Set2,Set1}. So, there are total 6 combination of selecting the two sets.
The above is an example of combinations. Selection of r things that can be done out of total n things is defined as Combinations. This is denoted by ^{n}C_{r} which is equal to n! / r! (n-r)! where 0r n.
Note : Permutations of n different things taken r at a time is ? ^{n}?P_{?r} ? = ? ^{n?}C?_{r }? r!or ? ^{n}?C?_{r?} = ? ^{n}?P_{?r} ? / r!
^{n}?C_{?r - 1} ?+ ? ^{n}?C_{?r}? = ? ^{n+1?}C_{?r}
Note:? It is difficult to identify the problem type in Permutations and Combinations questions. Sometimes you can mistakenly approach a permutation problem as a combination, and vice versa. So it is very necessary to recognise the problem type i.e permutation or combination or both. You can use formulas or models to count the possibilities
Importance of Permutations and Combinations:
Permutations and Combinations are also known as PnC is that tool in the field of mathematics that we have been using since our childhood. In our everyday’s life while taking any decision we try to look for options and forms possible combinations or arrangement to make any decision. Permutation and Combinations become very handy in making choices in our daily life. Apart from our daily life, PnC is frequently used in communication networks and parallel and distributed systems. In the field of computer architecture designing of computer chips involves permutation and combinations of input to output pins. Computational molecular biology involves many types of combinatorial and sequencing problems such as atoms, molecules, DNAs etc. Study of permutation and combination plays an important role in pattern analysis, for scientific discovery problems. In the field of databases and data mining, queries are a permutation of the join operations, for example, determination of an optimal permutation that gives minimum cost is a common and important problem. PnC is also used for simulations in many areas such as cryptography and network security
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