Lecture on the topic: Combinatorics

these are tuples in which the element is repeated once.

Example. How many different "words" can be formed by rearranging the letters of the word "mom"?

Solution:

There are 4 letters in the word "mom": n = 4

The letter "m" occurs in the word 2 times:

The letter "a" - 2 times: . According to the formula we get:

Combinations

Combinations of n elements by m elements are combinations made up of given n elements by m elements that differ by at least one element .

The difference between combinations and placements is that combinations do not take into account the order of elements. The number of all combinations of n elements by m elements is denoted by the symbol: . Combinations without repetition ( n different elements, taken m by m ):

Example. To conduct the exam, a commission of two teachers is created. How many different committees can be made if there are five teachers?

Example. To conduct the exam, a commission of two teachers is created. How many different committees can be made if there are five teachers?

n =5 , m=2

Various unordered sets composed of m elements of this set so that the elements in the set can be repeated, and their order is not important, are called combinations with repetitions from n to m. Their number is:

Example. Take the fruits banana, pineapple, kiwi, apple and turnip. What combinations of these fruits, taken two by two, can be obtained? How many such sets will there be if

Example. Take the fruits banana, pineapple, kiwi, apple and turnip. What combinations of these fruits, taken two by two, can be obtained? How many such sets will there be if

1) the fruits in the set are not repeated;

2) Can I take two identical fruits?