Question

determining the number of possible groups of elements
ten students need to present their reports. five can present each day. how many ways can the teacher choose a group of five students to present their reports on the first day?
how many ways can the teacher choose a group of 5 students to present on the first day if marjorie must present on the first day?
126
252
3,024
30,240

Explanation:

Step1: Solve the first question (no restriction)

We need to choose 5 students out of 10, and the order does not matter (since a group is just a set of students), so we use the combination formula \( C(n, k)=\frac{n!}{k!(n - k)!} \), where \( n = 10 \) and \( k = 5 \).

First, calculate the factorials:
\( 10! = 10\times9\times8\times7\times6\times5! \)
\( 5! = 5\times4\times3\times2\times1 \)
\( (10 - 5)! = 5! \)

Then, substitute into the combination formula:
\( C(10, 5)=\frac{10!}{5!5!}=\frac{10\times9\times8\times7\times6\times5!}{5!\times5\times4\times3\times2\times1} \)

Cancel out the \( 5! \) terms:
\( C(10, 5)=\frac{10\times9\times8\times7\times6}{5\times4\times3\times2\times1} \)

Calculate the numerator: \( 10\times9\times8\times7\times6 = 30240 \)
Calculate the denominator: \( 5\times4\times3\times2\times1 = 120 \)
Then, \( \frac{30240}{120}=252 \)

Step2: Solve the second question (Marjorie must be in the group)

If Marjorie must be in the group, we already have 1 student (Marjorie) in the group, so we need to choose \( 5 - 1 = 4 \) more students from the remaining \( 10 - 1 = 9 \) students.

Using the combination formula again, with \( n = 9 \) and \( k = 4 \):
\( C(9, 4)=\frac{9!}{4!(9 - 4)!}=\frac{9!}{4!5!} \)

Calculate the factorials:
\( 9! = 9\times8\times7\times6\times5! \)
\( 4! = 4\times3\times2\times1 \)
\( 5! = 5\times4\times3\times2\times1 \)

Substitute into the formula:
\( C(9, 4)=\frac{9\times8\times7\times6\times5!}{4!\times5!} \)

Cancel out the \( 5! \) terms:
\( C(9, 4)=\frac{9\times8\times7\times6}{4\times3\times2\times1} \)

Calculate the numerator: \( 9\times8\times7\times6 = 3024 \)
Calculate the denominator: \( 4\times3\times2\times1 = 24 \)
Then, \( \frac{3024}{24}=126 \)

Answer:

First question: 252
Second question: 126 (the option with 126)