![]() ![]() The company has so far produced 500,000 cars, of which 5,000 were defective. How many ways can we choose four different meals for the daily menu? The probability (rounded to three decimal places) that the first adult possesses a credit card and the second adult does not possess a credit card is A researcher selects two adults at random. The probability that an adult possesses a credit card is 0.71. Are the random events A - the product has an appearance defect and B - the product has a functiona The product has a 10% probability of an appearance defect, a 6% probability of a functional deficiency, and a 3% probability of both defects simultaneously. What is the probability that a girl will be drawn? The circle leader wants to determine by lot which member will represent the circle at the representative tournament. The chess club has 5 members, including two girls. How many options do we have?įoundation of combinatorics in word problems k is logically greater than n (otherwise, we would get ordinary combinations).Ĭ k ′ ( n ) = ( k n + k − 1 ) = k ! ( n − 1 ) ! ( n + k − 1 ) ! Įxplanation of the formula - the number of combinations with repetition is equal to the number of locations of n − 1 separators on n-1 + k places.Ī typical example is: we go to the store to buy 6 chocolates. Here we select k element groups from n elements, regardless of the order, and the elements can be repeated. Their number is a combination number and is calculated as follows:Ĭ k ( n ) = ( k n ) = k ! ( n − k ) ! n ! Ī typical example of combinations is that we have 15 students and we have to choose three. ![]() In mathematics, disordered groups are called sets and subsets. The elements are not repeated, and it does not matter the order of the group's elements. k m ! n ! Ī typical example is to find out how many seven-digit numbers formed from the numbers 2,2,2, 6,6,6,6.Ī combination of a k-th class of n elements is an unordered k-element group formed from a set of n elements. Repeating some (or all in a group) reduces the number of such repeating permutations. n = n kĪ repeating permutation is an arranged k-element group of n-elements, with some elements repeating in a group. We calculate their number according to the combinatorial rule of the product: A typical example is the formation of numbers from the numbers 2,3,4,5, and finding their number. 1 = n !Ī typical example is: We have 4 books, and in how many ways can we arrange them side by side on a shelf?Ī variation of the k-th class of n elements is an ordered k-element group formed of a set of n elements, wherein the elements can be repeated and depends on their order. The elements are not repeated and depend on the order of the elements in the group. It is thus any n-element ordered group formed of n-elements. The permutation is a synonymous name for a variation of the nth class of n-elements. For calculations, it is fully sufficient to use the procedure resulting from the combinatorial rule of product. The notation with the factorial is only clearer and equivalent. ![]() N! we call the factorial of the number n, which is the product of the first n natural numbers. For example, if we have the set n = 5 numbers 1,2,3,4,5, and we have to make third-class variations, their V 3 (5) = 5 * 4 * 3 = 60. The number of variations can be easily calculated using the combinatorial rule of product. The elements are not repeated and depend on the order of the group's elements (therefore arranged). C k ( n ) = ( k n ) = k ! ( n − k ) ! n ! n = 1 0 k = 4 C 4 ( 1 0 ) = ( 4 1 0 ) = 4 ! ( 1 0 − 4 ) ! 1 0 ! = 4 ⋅ 3 ⋅ 2 ⋅ 1 1 0 ⋅ 9 ⋅ 8 ⋅ 7 = 2 1 0 The number of combinations: 210Ī bit of theory - the foundation of combinatorics VariationsĪ variation of the k-th class of n elements is an ordered k-element group formed from a set of n elements. ![]()
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