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Rules of Permutation and Combination
Permutation
Permutation is an ordered arrangement of a number of elements of a set. Mathematically, given a set with n numbers of elements, the number of permutations of size r is denoted by P(n,r) or nPr or nPr. The formula is given by P(n,r) = nPr = nPr = n! (n  r)! where n! (n factorial) = n × (n1) × (n2) × ... × 1 and 0! = 1. For example, given the set of letters {a,b,c} the permutations of size 2 (take 2 elements of the set) are {a,b}, {b,a}, {a,c}, {c,a}, {b,c}, and {c,b}. Please note that the order is important (i.e. {a,b} is considered different from {b,a}). The number of permutations is 6. P(3,2) = 3P2 = 3P2 = 3! (3  2)! = 3 × 2 × 1 1! = 6 1 = 6 Another example: How many different ways are there can 5 different books be arranged on the self? Answer: Here, n = 5 and r = 5. So, 5P5 = 5!/(55)! = 5!/0! = (5 × 4 × 3 × 2 × 1)/1 = 120. As can be seen from the above example, when n = r, the formula for nPr = n!.  Combination Combination is an unordered arrangement of a number of elements of a set. Given a set with n numbers of elements, the number of combinations of size r is denoted by C(n,r) or nCr or nCr. The formula is given by C(n,r) = nCr = nCr = n! r! (n  r)! where n! (n factorial) = n × (n1) × (n2) × ... × 1 and 0! = 1. For example, given the set of letters {a,b,c} the combinations of size 2 (take 2 elements of the set) are {a,b}, {a,c}, and {b,c}. Please note that the order is not important (i.e. {b,b} is considered the same as {a,b}). The number of combinations is 3. C(3,2) = 3C2 = 3C2 = 3! 2! (3  2)! = 3 × 2 × 1 2 × 1 × 1! = 6 2 = 3 Another example: A basket contains an apple, an orange, a pear, and a banana. How many combinations of three fruits are there? Answer: Here, n = 4 and r = 3. So, 5C5 = 4!/3!(43)! = (4 × 3 × 2 × 1)/(3 × 2 × 1) 1! = 24/6 = 4. For combination when n = r, the number of combinations is always equal to 1. 
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safdarrao (Saturday, July 10, 2010) 
#2




In the formulae aforementioned for Permutation & Combination , the sign of division '/' is either missing or I'm unable to see it, at least. In my humble opinion, correct formulae are as follows :
P(n,r) = nPr = n! / (nr)! and C(n,r) = nCr = n! / r! (nr)! 
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