Combinations without repetition n=7, k=1 result

Find out how many different ways you can choose k items from n items set. With/without repetition, with/without order.

Elements to choose from:

(n)

Elements chosen:

(k)

Is order important:

YesNo

Repetition allowed:

YesNo

Calculate

A bit of theory - the foundation of combinatorics

Variations

A 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.

$V_k(n)=n(n-1)(n-2)...(n-k+1)=\frac{n!}{(n-k)!}$

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 and equivalent. For calculations, it is fully sufficient to use the procedure resulting from the combinatorial rule of product.

Permutations

The 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 repetition

A 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:

$V_k^{\prime }(n)=n\cdot n\cdot n\cdot n...n=n^k$

Permutations with repeat

A 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.

$P_{k_1{k}_2{k}_3...k_m}^{\prime }(n)=\frac{n!}{k_1!k_2!k_3!...k_m!}$

A typical example is to find out how many seven-digit numbers formed from the numbers 2,2,2, 6,6,6,6.

Combinations

A 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:

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

A typical example of combinations is that we have 15 students and we have to choose three. How many will there be?

Combinations with repeat

Here 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:

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

Explanation of the formula - the number of combinations with repetition 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

• Chess
  How many ways can you select 4 fields on a classic chessboard with 64 fields so that fields don't have the same color?
• Rectangles
  How many rectangles with area 8855 cm² whose sides are natural numbers?
• Pizza
  A school survey found that 10 out of 15 students like pizza. If 5 students are chosen randomly, what is the probability that all 5 students like pizza?
• Count of triangles
  On each side of an ABCD square is 10 internal points. Determine the number of triangles with vertices at these points.

• Probabilities
  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. intersect and opposite and its combinations):
• Combinatorics
  The city has 7 fountains. Works only 6. How many options are there that can squirt?
• Chess
  How many different ways can you initiate a game of chess (first pass)?
• Three digits number
  From the numbers 1, 2, 3, 4, and 5, create three-digit numbers whose digits do not repeat, and the number is divisible by 2. How many numbers are there?
• Intersection of the lines
  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)?

• Medicine
  We test medicine on six patients. For all, the drug doesn't work. If the drug success rate of 20%, what is the probability that medicine does not work?
• Eight blocks
  Dana had the task of saving the eight blocks of these rules: 1. Between two red cubes must be a different color. 2. Between two blue must be two different colors. 3. Between two green must be three different colors. 4. Between two yellow blocks must be fo
• Permutations without repetition
  From how many elements can we create 720 permutations without repetition?
• Two-digit 3456
  Write all the two-digit numbers that can be composed of the digit 7,8,9 without repeating the digits. Which ones are divisible b) two, c) three d) six?
• Points in plane
  The plane is given 12 points, 5 of which are located on a straight line. How many different lines could be drawn from these points?

• Three-digit code
  The five cards with the numbers 1, 2, 3, 4, and 5 put together all three-digit odd numbers. How many are there?

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