Question

How many different three-member teams can be formed from six students?

Answer 1

Hint: In this question we have to find the number of different teams can be formed from given number of total students, for this we will the combination formula which is given by Number of combinations when ‘$r$’ elements are selected out of a total of ‘$n$’ elements is${}^n{C}_r={\displaystyle \frac{n!}{r!\left(n-r\right)!}}$, which can also be represented by ${}^n{C}_r={}^n{C}_{n-r}$.
, and by substituting the values in the formula we will get the required result.

Complete step by step answer:
Given that we need to choose 3 students from a group of 6 students. Since no two students are the same, you will need to determine the number of combinations.
We will use the combination formula, which is given by number of combinations when ‘$r$’ elements are selected out of a total of ‘$n$’ elements is given by ${}^n{C}_r={\displaystyle \frac{n!}{r!\left(n-r\right)!}}$,,
So, here where $n$ is the number of students total and r is the number of students that need to be chosen.
Total number of students = $n$=6
Number of students that we need to select $r$= 3,
By substituting the values in the formula we get,
$\Rightarrow {}^6{C}_3={\displaystyle \frac{6!}{3!\left(6-3\right)!}}$,
Now simplifying we get,
$\Rightarrow {}^6{C}_3={\displaystyle \frac{6!}{3!\left(3!\right)}}$,
Now again simplifying we get,
$\Rightarrow {}^6{C}_3={\displaystyle \frac{6\times 5\times 4\times 3\times 2\times 1}{\left(3\times 2\times 1\right)\left(3\times 2\times 1\right)}}$,
Now eliminating the like terms we get,
$\Rightarrow {}^6{C}_3={\displaystyle \frac{6\times 5\times 4}{\left(3\times 2\times 1\right)}}$,
Again simplifying we get,
$\Rightarrow {}^6{C}_3=20$,
So, 3 team members from 6 students can be formed in 20 ways.

$\therefore$ There are 20 ways to choose 3 students from a group of 6 students.

Note: Combination is the different selections of a given number of elements taken one by one, or some, or all at a time. For example, if we have two elements A and B, then there is only one way to select two items, we select both of them. As the question is related to combinations, we should know the definition and the formula related to the combinations and students should understand the question, and the condition given, as they may get confused in finding the arrangements, which should be done according to the condition given in the question.