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

• STRESSED word
  Each letter in STRESSED is printed on identical cards, one letter per card, and assembled in random order. Calculate the probability that the cards spell DESSERTS when assembled.
• Fruits
  We want to plant five fruit trees in the garden, of which three are apple trees and two pears. How many different ways can we organize them?
• Digits
  How many odd four-digit numbers can we create from digits 0, 3, 5, 6, and 7? (a) the figures may be repeated (b) the digits may not be repeated
• Families 2
  Seven hundred twenty-nine families have six children each. The probability of a girl is 1/3, and the likelihood of a boy is 2/3. Find the number of families having two girls and four boys.
• Two groups
  The group of 10 girls should be divided into two groups with at least four girls in each group. How many ways can this be done?

more math problems »