What is the probability that the six will fall exactly four times?

How many different options do you have?ĭetermine the number of all four-digit natural numbers in decimal notation in which the digit 0 is not present, and each of the remaining nine numbers occurs at most once.

For dessert, you have a choice of ice cream and mango peach. In a Jollibee, you have a menu choice of C1, C2, and C3. How many points do nine lines intersect in a plane, of which four are parallel, and of the other five, no two are parallel (and if we assume that only two lines pass through each intersection)? How many ways can we select five items so that it is no more than one defective? 36 are first-grade, and four are defective. What is the probability that we will have to make a fifth trial? Attempts are repeated until this phenomenon occurs. The probability of occurrence of a certain phenomenon is the same in all trials and is equal to 0.7.

intersect and opposite and its combinations): If probabilities of A, B, and A ∩ B are P (A) = 0.62, P (B) = 0.78, and P (A ∩ B) = 0.26, calculate the following probability (of the union. How many ways can a teacher select a group of 6 students to sit in the front row if the class has 13 students? What is the probability that it gets a) all four aces b) at least one ace How many options do we have?įoundation of combinatorics in word problemsĬalculate: (486 choose 159) - (486 choose 327) 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). A 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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