Elements of combinatorics. Combinatorics

check yourself

2) In how many ways can a set of white pieces (a king, a queen, two rooks, two bishops and two knights) be placed on the first horizontal of the chessboard?
Solution.
A set of white chess pieces consists of 8 pieces:
1 king, 1 queen, 2 rooks, 2 bishops and 2 knights
(m= 8; k1, k2, … , kn)
R 2,3,2,1,1,1 =
8 !
1! 1! 2! 2! 2!
=
5040.
3*4*5*6*7*8
2*2
=

check yourself

3) Mom has 2 apples, 3 pears and 4 oranges. Every day for nine consecutive days, she gives her son one of the remaining fruits. In how many ways can this be done?
SOLUTION

check yourself

3) Mom has two apples, three pears and four oranges. Every day for nine consecutive days, she gives her son one of the remaining fruits. In how many ways can this be done?
Solution.
Mom has only 9 fruits: two apples, three pears and four oranges. (k 1 , k 2 , … , k n )
R 2,3,4 =
9 !
2! 3! 4!
=
1260.
5*6*7*8*9
2*2*3
=

Historical reference

Combinatorial motifs can be seen even in the symbolism of the Chinese "Book of Changes" (5th century BC).

In the XII century. Indian mathematician Bhaskara in his main work " Lilavati " studied in detail problems with permutations and combinations, including permutations with repetitions.

Accommodations

Accommodation out of n elements by k

( ) is any set consisting of any k elements taken in a certain order from n elements.

Two arrangements of n elements are considered different if they differ in the elements themselves or in the order and x of the arrangement.

Example

In how many ways can an asset be identified out of 40 students in the following composition: the headman, the fizorg and the editor of the wall newspaper?

Solution: