Question

How many subsets of four elements can be formed from a set of 100 elements?

Answer 1

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${}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$, 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 four elements can be formed from a set of 100 elements. Since no two numbers are the same, we 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${}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$,
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 elements = $n = 100$,
Number of subsets that we need to select $r$= 4,
By substituting the values in the formula we get,
$\Rightarrow {}^{100}{C_4} = \dfrac{{100!}}{{4!\left( {100 - 4} \right)!}}$,
Now simplifying we get,
$\Rightarrow {}^{100}{C_4} = \dfrac{{100!}}{{4!\left( {96} \right)!}}$,
Now again simplifying we get,
$\Rightarrow {}^{100}{C_4} = \dfrac{{100 \times 99 \times 98 \times 97 \times 96!}}{{\left( {4 \times 3 \times 2 \times 1} \right)\left( {96} \right)!}}$,
Now eliminating the like terms we get,
$\Rightarrow {}^{100}{C_4} = \dfrac{{100 \times 99 \times 98 \times 97}}{{\left( {4 \times 3 \times 2 \times 1} \right)}}$,
Again simplifying we get,
$\Rightarrow {}^{100}{C_4} = \dfrac{{94109400}}{{24}}$,
Further simplification we get,
$\Rightarrow {}^{100}{C_4} = 3921225$,

So, 4 subsets from a total of 100 elements can be formed in 3921225 ways.

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.