Combinations with repetitionFind out how many different ways you can choose k items from n items set. With/without repetition, with/without order.
Ck′(n)=(kn+k−1) n=11 k=3 C3′(11)=C3(11+3−1)=C3(13)=(313)=3!(13−3)!13!=3⋅2⋅113⋅12⋅11=286
Number of combinations with repetition: 286
A bit of theory - foundation of combinatorics
VariationsA variation of the k-th class of n elements is an ordered k-element group formed from a set of n elements. The elements are not repeated and depend on the order of the group's elements (therefore arranged).
The number of variations can be easily calculated using the combinatorial rule of product. For example, if we have the set n = 5 numbers 1,2,3,4,5 and we have to make third-class variations, their V3 (5) = 5 * 4 * 3 = 60.
Vk(n)=n(n−1)(n−2)...(n−k+1)=(n−k)!n!n! we call the factorial of the number n, which is the product of the first n natural numbers. The notation with the factorial is only clearer, equivalent. For calculations, it is fully sufficient to use the procedure resulting from the combinatorial rule of product.
PermutationsThe permutation is a synonymous name for a variation of the nth class of n-elements. It is thus any n-element ordered group formed of n-elements. The elements are not repeated and depend on the order of the elements in the group.
P(n)=n(n−1)(n−2)...1=n!A typical example is: We have 4 books, and in how many ways can we arrange them side by side on a shelf?
Variations with repetitionA 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. A typical example is the formation of numbers from the numbers 2,3,4,5, and finding their number. We calculate their number according to the combinatorial rule of the product:
Permutations with repeatA repeating permutation is an arranged k-element group of n-elements, with some elements repeating in a group. Repeating some (or all in a group) reduces the number of such repeating permutations.
Pk1k2k3...km′(n)=k1!k2!k3!...km!n!A typical example is to find out how many seven-digit numbers formed from the numbers 2,2,2, 6,6,6,6.
CombinationsA combination of a k-th class of n elements is an unordered k-element group formed from a set of n elements. The elements are not repeated, and it does not matter the order of the group's elements. In mathematics, disordered groups are called sets and subsets. Their number is a combination number and is calculated as follows:
Ck(n)=(kn)=k!(n−k)!n!A typical example of combinations is that we have 15 students and we have to choose three. How many will there be?
Combinations with repeatHere we select k element groups from n elements, regardless of the order, and the elements can be repeated. k is logically greater than n (otherwise, we would get ordinary combinations). Their count is:
Ck′(n)=(kn+k−1)=k!(n−1)!(n+k−1)!Explanation 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. A typical example is: we go to the store to buy 6 chocolates. They offer only 3 species. How many options do we have? k = 6, n = 3.
Foundation of combinatorics in word problems
- Calculation of CN
Calculate: (789 choose 786) - (789 choose 3)
- Soccer teams
Have to organize soccer teams. There are 3 age groups. How many different ways can you organize teams of ten for each age group? Is this a permutation or combination?
How many 4-tones chords (chord = at the same time sounding different tones) is possible to play within 7 tones?
The player gets 8 cards of 32. What is the probability that it gets a) all 4 aces b) at least 1 ace
- Permutations without repetition
From how many elements, we can create 720 permutations without repetition?
- First class
The shipment contains 40 items. 36 are first grade, 4 are defective. How many ways can select 5 items, so that it is no more than one defective?
- School parliament
There are 18 boys and 14 girls in the class. In how many ways can 3 representatives be elected to the school parliament, if these are to be: a) the boys themselves b) one boy and two girls
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 union. intersect and opposite and its combinations):
- Pins 2
how many different possible 4 digits pins can be found on the 10-digit keypad?
The hockey match ended 8:2. How many different matches could be?
- Cancel fractions
Compress the expression of factorial: (n+6)!/(n+4)!-n!/(n-2)!
- Face combinations
If I have 20 sets of eyes, 20 noses, and 20 mouths, how many unique face combinations can I make?
We test medicine on 6 patients. For all drug doesn't work. If the drug success rate of 20%, what is the probability that medicine does not work?
From the bag with numbered balls (numbers 1,2,3,. ..20) we pick one ball. What is the probability of choosing a number containing 1?
C(6,3) + 3 P(6,3)
- One dice
Calculate the probability of one dice roll with the numbers 1, 2, 3, 4, 5, 6 on the walls. Write the results in a notebook in the shape of a fraction in the basic form: 2/3. a, The number 1 falls on the cube. b, The number 5 falls on the cube. c, An even
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