Combinations with repetition

The calculator finds the number of combinations of the k-th class from n elements with repetition. A combination with repetition of k objects from n is a way of selecting k objects from a list of n. The order of selection does not matter and each object can be selected more than once (repeated).

Elements to choose from:

(n)

Elements chosen:

(k)

Calculate

A bit of theory - the foundation of combinatorics

Combinations with repetition

Here we select k-element groups from n elements, regardless of order, where elements can be repeated. k is generally greater than n (otherwise we would obtain ordinary combinations). The count is:

$C_k^{\prime }(n)=\left(\genfrac{}{}{0ex}{}{n+k-1}{k}\right)=\frac{(n+k-1)!}{k!(n-1)!}$

Explanation of the formula: the number of combinations with repetition equals the number of ways to place n − 1 separators among n − 1 + k positions. A typical example: we go to a store to buy 6 chocolates. There are only 3 types available. How many different selections can we make? k = 6, n = 3.

Foundation of combinatorics in word problems

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• Syrups
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• Six attractions
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• Three workplaces
  How many ways can we divide nine workers into three workplaces if they need four workers in the first workplace, 3 in the second workplace, and 2 in the third?
• Beads
  How many ways can we thread four red, five blue, and six yellow beads onto a thread?
• Dice Rolling Six Probability
  Three players roll the dice. They can start the game when a six is rolled. Everyone rolls 1 time. a) What is the probability that exactly one starts in the first round? b) What is the probability that at least two start?
• Play match
  A hockey match played for three periods ended with a score of 2:3. How many possibilities are there on how the given thirds could have been completed?
• Grouping - combinatorics
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• Salami
  We have six kinds of salami, six of which have ten pieces, and one of which has four pieces. How many ways can we distinctly choose five pieces of salami?
• Bouquets
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• Wagons
  We have six wagons: two white, two blue, and two red. We assemble trains from them; wagons of the same color are exactly the same, so if we change only two white wagons on a train, it's still the same train because I don't know any difference. How many di
• Candy flavors
  A bag contains 20 candies in five different flavors: cherry, lemon, orange, mango, and cola. We know that there is at least one of each flavor in the pocket and that there are twice as many lemons as cherry ones. How many ways can different flavors be rep
• Card drawing
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• Three-digit numbers
  How many three-digit numbers do we make from the numbers 4,5,6,7?

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