Question

From a class of 32 students, 4 are to be chosen for a competition. Find in how many ways can this be done?

Answer 1

Use the formula of combinations given as ${}^{n}{{C}_{r}}$ to find the numbers of possible ways to select r things out of a total of n things. Consider n as the total number of students in the class and r as the number of students required to be selected for the competition. Now, use the formula $^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}$ to simplify the expression and get the answer in number.

Complete step by step answer:
Here we have been given that a class has 32 students in which 4 students will be selected for a competition. We are asked to determine the total number of ways in which the selection can be done.
Now, we know that if we have to select r number of things from a total of n number of things then we use the formula of combination given as ${}^{n}{{C}_{r}}$. So let us consider the total number of students as n and the number of students to be selected for the competition as r, so we have n = 32 and r = 4, substituting these values in the formula we get,
$\Rightarrow$ Number of possible ways of selection = ${}^{n}{{C}_{r}}={}^{32}{{C}_{4}}$
Using the formula $^{n}{{C}_{r}}=\frac{n!}{r!\left( n-r \right)!}$ we get,
$\Rightarrow$ Number of possible ways of selection = $\dfrac{32!}{4!\left( 32-4 \right)!}$
$\Rightarrow$ Number of possible ways of selection = $\dfrac{32!}{4!28!}$
We can write $n!=n\left( n-1 \right)!$, so we get,
$\Rightarrow$ Number of possible ways of selection = $\dfrac{32\times 31\times 30\times 29\times 28!}{4!28!}$
$\Rightarrow$ Number of possible ways of selection = $\dfrac{32\times 31\times 30\times 29}{4\times 3\times 2\times 1}=35960$

Hence, the total number of ways to select the required number of students is 35960.

Note: Do not use the formula of permutation because we are not asked to arrange these students but we have to select them only. You must know the difference between the two terms ‘permutation’ and ‘combination’. In general if we have to select r things from n thing then we apply the combinations formula and if after selection we need to arrange those things also then we apply permutation formula given as ${}^{n}{{P}_{r}}=\dfrac{n!}{\left( n-r \right)!}$.