Math Problem Statement

Soit E un ensemble de 8 éléments. Alors le nombre de parties de E composées de 2 éléments est égal à : 2. Dix équipes de basketball participent à un tournoi. Chaque équipe doit rencontrer toutes les autres une seule fois. Le nombre de matches à organiser est : 3. Voici la sixième ligne du triangle de Pascal. Sur la ligne suivante, le nombre situé dans la 4e case en partant de la gauche est égal à :

Solution

Let's break down the questions from the image you provided:

  1. Question 1: We need to find the number of pairs of 2 elements that can be formed from a set of 8 elements.

    This is a combination problem, and the formula for combinations is:

    C(n,r)=n!r!(nr)!C(n, r) = \frac{n!}{r!(n - r)!}

    For this case, we have n=8n = 8 and r=2r = 2, so:

    C(8,2)=8!2!(82)!=8×72×1=28C(8, 2) = \frac{8!}{2!(8 - 2)!} = \frac{8 \times 7}{2 \times 1} = 28

    So, the answer to the first question is d. 28.

  2. Question 2: There are 10 basketball teams, and each team plays against all the others exactly once. This can be calculated as the number of ways to choose 2 teams out of 10, which is also a combination problem:

    C(10,2)=10!2!(102)!=10×92×1=45C(10, 2) = \frac{10!}{2!(10 - 2)!} = \frac{10 \times 9}{2 \times 1} = 45

    The answer to the second question is d. 45.

  3. Question 3: This is asking for a number from the next line of Pascal's Triangle, based on the 6th row provided in the image. The numbers from the 6th row are:

    1,5,10,10,5,11, 5, 10, 10, 5, 1

    The question asks for the number in the 4th position (counting from the left), which is 10.

    So the answer to the third question is c. 10.

Would you like more details on any of these questions or need further clarifications? Here are some related questions you can explore next:

  1. How do we calculate combinations for larger sets (e.g., C(12,4)C(12, 4))?
  2. What is the difference between combinations and permutations?
  3. How is Pascal's Triangle useful in binomial expansions?
  4. Can we apply the combination formula to problems involving objects that are not distinct?
  5. How do combinations relate to the number of ways to form a committee or team?

Tip: Always remember that combinations involve selecting items without regard to order, while permutations do take order into account.

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Math Problem Analysis

Mathematical Concepts

Combinations
Pascal's Triangle

Formulas

Combination formula: C(n, r) = n! / (r!(n - r)!)
Pascal's Triangle properties

Theorems

Combination theorem
Pascal's Triangle row identification

Suitable Grade Level

Grades 7-9