Elements of combinatorics. Combinatorics

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3. In a store where there are 4 types of balls , they decided to put 8 balls in a row . In how many ways can this be done if their location matters?

Solution.

4 different types of balls, 8 positions, i.e. the number of different placements will be = 65536.

Answer : 65536 ways.

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six buttons of one of the four colors be sewn on a clown costume in a line to get a pattern?

SOLUTION

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• six buttons of one of the four colors be sewn on a clown costume in a line to get a pattern?

Solution.

Apparently, the number of buttons of each type is large, so to determine the number of ways, you can use the placement formula with repetitions.

It is equal to = 1296 (6 positions and 4 types).

Answer: 1296 ways.

Combinations

Combinations - compounds containing m items out of n , differing from each other in at least one item .

Combinations are finite sets in which the order does not matter.

Combinations

The formula for finding the number of combinations without repetitions:

Historical reference
In 1666, Leibniz published Discourses on Combinatorial Art. In his work, Leibniz, introducing special symbols, terms for subsets and operations on them, finds all k -combinations of n elements, deduces the properties of combinations:
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Usage example:
In how many ways can two attendants be chosen from a class of 25 students?
m = 2 ( required number of attendants )
n = 25 (total students in the class)
Solution:
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1) In how many ways can three students be delegated to an interuniversity conference of 9 members of a scientific society?
SOLUTION
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