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Permutations and combinations have been around since the beginning of Jainism in India, and possibly much before that. The credit, however, goes to the Jains, who, under the term Vikalpa, handled the subject matter as a self-contained issue in mathematics. Selections are another term for combinations. Combinations refer to the selecting of items from a set of options. We have no intention of arranging things here. We plan to choose them. Out of a set of n objects, we count the number of unique r-selections or combinations. nCr.
When simply the number of feasible groups needs to be identified and the order/sequence of arrangements isn’t important, combinations are employed. Let us learn combinations in detail.
Combination
The ways of selecting or choosing different person or object out of the total objects are person taking one or more than one at a time is called combination. Let r = different objects, n= total objects.
nCr = nỊ / (n-r)ỊrỊ
nCr is a combination of ‘r’ objects out of total ‘n’ objects.
If you have to select one or more than one subject at a time from four subjects then it is given by
selection of one out of four = 4C1 = 4Ị/3Ị1Ị = 4
selection of two out of four = 4C2 = 4Ị/2Ị2Ị = 4*3/2 = 6
selection of three out of four = 4C3 =4Ị/1Ị3Ị = 4
selection of four out of four = 4C4 =4Ị/0Ị4Ị = 1
Formula of combination
Combinations are calculated using the combinations formula, as well as factorials and permutations. In general, imagine we have n items at our disposal and we wish to determine the number of ways we can choose r items from these n items. We begin by calculating the total number of permutations of these n items taken r at a time. n P r would be that number. Since r items can be permuted amongst themselves in r! ways, each combination in this list of n P r permutations will be enumerated r! times. Thus, n C r denotes the entire number of permutations and combinations of these n items, taken r at a time.
Combination: nCr = nỊ / (n-r)ỊrỊ
Types of questions
1. Simple questions involving the selection of an object from a set of objects without any conditions or based on the counting concept.
2. Using n points, make a number of triangles, line squares, and parallelograms.
3. Distribution of various goods to various persons
4. The design is based on a circular pattern.
Examples of combination
Type 1- The number of ways of selecting 6 players out of 11 players.
Sol- n = 11 , r = 6 (given)
∴ number of ways of selecting 6 players out of 11
11C6 = 11!/5!6!
= 11*10*9*8*7/5*4*3*2*1
=11*2*3*7 = 462.
Type2 – In a plane, there are 15 points, none of which are in a straight line except for 6, which are all in a straight line. The maximum number of straight lines that can be drawn in a single drawing.
Sol- Total number of lines if all 15 points are scattered.
However, six points are in a straight line. If the six points are also scattered, the number of lines generated by connecting these six spots increases.
6C2 = 6!/2! 4! = 6*5/2 = 15.
Total lines formed by 15 points in a plane out of which 6 are in straight line
15C2 – 6C2 +1 = 91
Some questions for practice
Q1. Out of 10 blue and 8 white balls, 5 blue balls and 4 white balls can be drawn in how many ways.
Q2. From a group of 7 men and 6 women, five persons are to be selected to form a committee so that at least 3 men are there in the committee. In how many ways can it be
Relationship between permutation and combination
The mathematics and principles of permutation and combination are very similar. Assume you have a total of n objects. You must calculate the number of distinct r-selections (selections containing r objects) that can be made from this collection of n objects. Consider a group of n people: you must determine the number of distinct subgroups of size r that can be formed from this group.
The number of size r permutations will be n P r. Because the objects in an r-selection can be permuted among themselves in r! ways, each unique selection will be counted r! times in the list of nPr permutations. As a result, the number of unique combinations can be as high as nP r/ r!
nCr = nPr/r!= n!/(n-r)r! = n!/r!(n-r)
Conclusion
Combination is the way of selecting items in which the order or sequence of selection doesn’t matter. Formula to calculate permutation is nPr = (n!) / (n-r)! and formula to calculate combination is nCr = nỊ / (n-r)ỊrỊ. 6Ị(6 factorial) means Multiply 6 in decreasing order until you get 1 i.e.6*5*4*3*2*1. We use permutation when we have to we have to arrange the objects and different orders are to be counted. Use combination if a problem is for the number of ways of selecting objects and the Order of selection is not important. Consider the phrase “number of options” whenever you hear the phrase “number of combinations.” It doesn’t matter what order the objects are in when you’re picking them. XYZ and XZY, for example, are two distinct arrangements with the same selection. The number of different combinations of n different things that can be made r at a time (where r is smaller than n) = nCr = nPr/r!= n!/(n-r)r! = n!/r!(n-r)
