Math Problem Statement

There are 5 girls and 4 boys in a classroom.How many 4 member  groups can be made

Question 14Answer

a.

67

b.

78

c.

126

d.

240

Solution

To solve this problem, we need to calculate the number of ways to form a 4-member group from a total of 9 students (5 girls + 4 boys).

The formula for combinations, which gives the number of ways to choose a subset of kk elements from a set of nn elements, is given by:

Number of combinations=(nk)=n!k!(nk)!\text{Number of combinations} = \binom{n}{k} = \frac{n!}{k!(n-k)!}

Here, n=9n = 9 (total students) and k=4k = 4 (group size). So, we have:

(94)=9!4!(94)!=9×8×7×64×3×2×1=302424=126\binom{9}{4} = \frac{9!}{4!(9-4)!} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = \frac{3024}{24} = 126

So, the correct answer is:

c. 126

Would you like more details or have any questions about this solution? Here are some additional related questions you might find useful:

  1. How many 3-member groups can be made from the same set of 9 students?
  2. What is the probability of randomly selecting a group of 4 students that includes exactly 2 girls?
  3. How many ways can we form a group of 4 students with at least 1 girl?
  4. How many different ways can we select 2 boys and 2 girls from the group?
  5. If we add 1 more girl to the classroom, how many 4-member groups can be made?

Tip: When calculating combinations, the order of selection doesn't matter. This distinguishes combinations from permutations, where order is important.

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Math Problem Analysis

Mathematical Concepts

Combinatorics
Permutations and Combinations

Formulas

Combination formula: \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \)

Theorems

Basic Principle of Counting

Suitable Grade Level

Grades 9-11