There are 10 students in a classroom including a class monitor. How many ways are there to make a group of 4 students In a way that the class leader is **not** included in a group.

Question 27Answer

a.

233

b.

126

c.

66

d.

86

To solve the problem of finding the number of ways to make a group of 4 students without including the class monitor, follow these steps:

1. **Identify the total number of students available for selection excluding the class monitor.**  
   Since the class monitor cannot be included, we have 9 students left (10 total - 1 class monitor).

2. **Calculate the number of ways to choose 4 students from the 9 remaining students.**  
   This is a combination problem, and the formula for combinations is given by:

   \[
   \text{Number of combinations} = \binom{n}{r} = \frac{n!}{r!(n-r)!}
   \]

   Here, \( n = 9 \) and \( r = 4 \).

   \[
   \binom{9}{4} = \frac{9!}{4!(9-4)!} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = \frac{3024}{24} = 126
   \]

Therefore, the number of ways to form a group of 4 students without including the class monitor is **126**.

The correct answer is **(b) 126**.

Would you like to see the details or have any questions? Here are some related questions you might find useful:

1. How do you calculate combinations when choosing multiple objects?
2. What is the difference between combinations and permutations?
3. How would the answer change if we wanted to include the class monitor in every group?
4. How many ways can you form a group if the group size was 5 instead of 4?
5. How would the problem change if there were two class monitors, and neither could be included?

**Tip:** Remember, combinations are used when the order of selection does not matter, whereas permutations are used when the order does matter.

There are 10 students in a classroom including a class monitor. How many ways are there to make a group of 4 students in a way that the class leader is not included in a group.

Discover how to calculate the number of ways to form a group of 4 students from a class of 10, ensuring the class monitor is not included. This problem involves key combinatorial concepts and is suitable for students in Grades 10-12.

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