On the other hand, combination focuses on the selection of objects from a given set without considering their order. It calculates the number of ways objects can be arranged, taking into account all the different positions each object can occupy. Permutation refers to the arrangement of objects in a specific order, considering every possible order or sequence. This concludes our disucssion on the topic of combination formula.To explain permutation and combination: Permutation and combination are mathematical concepts that deal with the arrangement and selection of elements. Therefore, the total number of ways = 5C 3 × 12C 5 × 8! = 319334400 ways. These eight letters can arrange themselves in 8! = 8P 8 ways. So, the number of ways of selecting 3 vowels and 5 consonants is 5C 3 × 12C 5. The number of ways of selecting 5 consonants from 12 consonants is 12C 5. Solution: The number of ways of selecting 3 vowels from 5 vowels is 5C 3. Problem: From a word containing 5 vowels and 12 consonants, how many 8 letter words can be formed by using 3 vowels and 5 consonants? Solution: We know that if nC p = nC q, then either p = q or p + q = n. So, the total number of ways of selections = (a + 1) (b + 1) (c + 1) 2 x – 1. But this includes the case of rejection of all the items. When the items are all different, the number of selecting x items is 2 x. Similarly, the number of ways of selecting b is (b + 1) and so on. Hence there are (a + 1) ways of selecting a. Out of ‘a’ items we can select 0, 1, 2, …, a items. Out of these items, ‘a’ are of the first kind, ‘b’ are of the second kind, ‘c’ are of the third kind and so on and the remaining x are all different. Suppose in a group of n items, there exist some objects which are of a similar kind and a few of them are different. So, the total number of ways of selection of one or more items = 2 n – 1. Therefore, the total number of ways for either situation is 2 × 2 × 2 × 2 … n times = 2 n. Here, each item is categorized in two ways i.e., it may get included or excluded. Let us consider the various situations of selecting r different items from n items when All Items are Different If nC k = nC r, then either k = r or k + r = n.Important Results Related to Combination Formula The relation between a permutation and a combination is If there are n number of items out of which we need to select r of them. We have in total of 12 ways of arranging the fruits. Since each of the two selected fruit can arrange themselves in 2! ways. Also after selecting the fruits if we arrange them in an order the above situation of combination reduces to that of a permutation. If we take into consideration the order in which we select a fruit first and then after the second one gives a situation of permutation. Relation Between Permutation and Combination FormulaĬonsider the above example of selecting two fruits from the four. We can say that there are six possible ways of selection of two fruits out of four. This selection of fruits does not depend upon the order of the selections. The symbols have their respective meanings. In how many ways can you select the two fruits? What are the possible selections? The possible selections are AG, AP, AO, GP, GO, and PO. You want to select any two fruits from them. Suppose you have four fruits Apple (A), Guava (G), Pear (P), and Orange (O). When the selected things are arranged in all possible orders, each of the arrangements is a permutation. A combination for selecting r items from n items is denoted by the symbol nC r. In combination, the selection or the collection of things is irrespective of the order of selection. In other words, each of the selections which can be made by taking some or all of the number of the things is a combination. The combination shows the number of the ways in which we can select different objects.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |