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?

Explanation:

Step1: Calculate the number of ways to choose 5 students out of 10

The formula for combinations is \(C(n,r)=\frac{n!}{r!(n - r)!}\), where \(n = 10\) and \(r=5\).

$$
LATEXBLOCK0
$$

Step2: Calculate the number of ways when Marjorie must be in the group

Since Marjorie is already in the group, we need to choose \(4\) more students out of the remaining \(9\) students. Using the combination formula \(C(n,r)\) with \(n = 9\) and \(r = 4\)

$$
LATEXBLOCK1
$$

Answer:

The number of ways to choose a group of 5 students out of 10 is \(252\). The number of ways when Marjorie must be in the group is \(126\).