Exercises on Multiplication and Addition Rules

This page contains several exercises applying the multiplication and addition rules. The problems range from simple counting tasks to more complex scenarios involving clothing combinations and number selections.

Highlight: Exercise 6/104 demonstrates the practical application of the multiplication rule in choosing outfits, where 7 shirts and 3 jackets result in 21 possible combinations.

The exercises progressively increase in complexity, helping students understand how to apply these rules in various situations. For instance, exercise 7/104 involves multiple steps and conditions, requiring a careful application of both the multiplication and addition rules.

Wariacje z Powtórzeniami

This page introduces the concept of wariacje z powtórzeniami (variations with repetition). It explains how to create sequences from a set of elements, allowing repetitions. The formula for calculating the number of variations with repetition is presented.

Example: The page demonstrates how to calculate the number of five-letter words (including nonsensical ones) that can be formed using the letters {A, B, C}. The answer is 3^5 = 243 possible words.

The page also includes exercises to practice this concept, such as calculating the number of possible routes and PIN codes under different conditions.

Wariacje bez Powtórzeń

This section covers wariacje bez powtórzeń (variations without repetition). It explains how to create sequences from a set of elements without allowing repetitions. The formula for calculating variations without repetition is provided.

Vocabulary: Wariacja bez powtórzeń refers to an arrangement of elements where each element can be used only once in the sequence.

The page includes an example of calculating the number of four-digit PIN codes composed of different digits. It also presents more complex exercises involving passenger arrangements in train wagons and number selections under specific conditions.

Permutacje

This page introduces the concept of permutations. It explains that a permutation is an n-element sequence consisting of all elements of a set. The formula for calculating the number of permutations is presented as P_n = n!.

Example: The page demonstrates how to calculate the number of ways to arrange 5 people in a queue, which is 5! = 120.

Several exercises are provided to practice calculating permutations in various scenarios, such as arranging people or objects in different orders.

Kombinacje

The final page covers the concept of combinations. It explains how to calculate the number of ways to choose k elements from an n-element set. The formula for combinations is presented as C(n,k) = n! / $k!(n-k)!$.

Definition: A kombinacja is a selection of items from a larger set where the order does not matter.

Two examples are provided to illustrate the application of the combination formula:

1. Calculating the number of ways to choose 2 people from a class of 30.
2. Determining the number of ways to select 3 players from a team of 12.

These examples help students understand how to apply the combination formula in practical scenarios.

Reguła Mnożenia i Dodawania

This page introduces two fundamental principles of combinatorics: the multiplication rule and the addition rule. The reguła mnożenia is explained using practical examples, such as selecting a delegation from a group of boys and girls, and creating meal combinations in a restaurant. The reguła dodawania is briefly introduced, explaining how to count elements from different sets.

Example: A restaurant offers 5 soups, 10 main courses, and 6 desserts. The total number of possible meal combinations is calculated as 5 × 10 × 6 = 300.

Definition: The reguła dodawania states that when choosing an element from two disjoint sets, the total number of choices is the sum of the elements in both sets.