Permutation and Combination are the ways to write a group of objects by selecting them in a specific order and forming their subsets. To arrange groups of data in a specific order permutation and combination formulas are used. Selecting the data or objects from a certain group is said to be permutations, whereas the order in which they are arranged is called combination. Permutation and Combination formulas are very useful in solving various problems in mathematics. Show
PermutationIt is the distinct interpretations of a provided number of components carried one by one, or some, or all at a time. For example, if we have two components A and B, then there are two likely performances, AB and BA. A numeral of permutations when ‘r’ components are positioned out of a total of ‘n’ components is n Pr = n! / (n – r)!. For example, let n = 3 (A, B, and C) and r = 2 (All permutations of size 2). The answer is 3!/(3 – 2)! = 6. The six permutations are AB, AC, BA, BC, CA, and CB. Explanation of Permutation Formula
CombinationIt is the distinct sections of a shared number of components carried one by one, or some, or all at a time. For example, if there are two components A and B, then there is only one way to select two things, select both of them. Number of combinations when ‘r’ components are chosen out of a total of ‘n’ components is, nCr = n! / [(r!) x (n – r)!]. For example, let n = 3 (A, B, and C) and r = 2 (All combinations of size 2). The answer is 3!/((3 – 2)! × 2!) = 3. The six combinations are AB, AC, and BC. nCr = nC(n – r)
Explanation of Combination Formula
Formulas Permutation and CombinationVarious formulas are used to solve permutation and combination problems few important of them are given below: Permutation FormulaPermutation formula is used to pick r things out of n different things in a specific order and replacement is not allowed. Combination FormulaCombination formula is used to pick r things out of n different things, where the order of picking is not important and replacement is not allowed. Derivation of Permutation and Combination FormulasPermutations Formula: Total number of permutations of a set of n objects if r objects are taken at a time P(n,r) = n!/(n-r)!, where [n>= r] Combination Formula: Total number of combinations of a set of n objects if r objects are taken at a time C(n,r) = n!/[r !(n-r)!], where [n>= r] Derivation of these formulas is discussed below in this article. Derivation of Permutations FormulaPermutation is selecting r distinct objects from n objects without replacement and where the order of selection is important, by the fundamental theorem of counting, we know P (n, r) = n . (n-1) . (n-2) . (n-3)…… (n-(r+1)) Multiplying and Dividing above by (n-r) (n-r-1) (n-r-2)……….. 3. 2. 1, P (n, r) = [n.(n−1).(n−2)….(nr+1)[(n−r)(n−r−1)(n−r−2)…3.2.1]/[(n−r)(n−r−1)(n−r−2)…3.2.1] P (n, r) = n!/(n−r)! Thus, the formula for P (n, r) is derived. Derivation of Combinations FormulaCombination is choosing r items out of n items when the order of selection is of no importance. Its formula is calculated as, C(n,r) = Total Number of Permutations /Number of ways to arrange r different objects. C(n,r) = P (n, r)/ r! C(n,r) = n!/(n−r)!r! Thus, the formula for P (n, r) is derived. Difference Between Permutation and CombinationVarious differences between the permutation and combination can be understood by the following points:
Example 1: Find the number of permutations and combinations of n = 9 and r = 3. Solution:
Example 2: In how many ways a committee consisting of 4 men and 2 women, can be chosen from 6 men and 5 women? Solution:
Example 3: How considerable words can be created by using 2 letters from the term“LOVE”? Solution:
Example 4: Out of 5 consonants and 3 vowels, how many words of 3 consonants and 2 vowels can be formed? Solution:
Example 5: How many different combinations do you get if you have 5 items and choose 4? Solution:
Example 6: Out of 6 consonants and 3 vowels, how many expressions of 2 consonants and 1 vowel can be created? Solution:
Example 7: In how many distinct forms can the letters of the term ‘PHONE’ be organized so that the vowels consistently come jointly? Solution:
FAQs on Permutations and CombinationsQuestion 1: What is the factorial formula? Answer:
Question 2: What does nCr represent? Answer:
Question 3: What do you mean by permutations and combinations? Answer:
Question 4: Write examples of permutations and combinations. Answer:
Question 5: Write the formula for finding permutations and combinations. Answer:
Question 6: Write some real-life examples of permutations and combinations. Answer:
Question 7: What is the value of 0!? Answer:
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