From a class of 25 students, 4 are to be chosen for a competition. In how many ways can this be done?
This is a case of combination:
Here
$\mathrm{n}=25$
$r=4$
$\Rightarrow{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}={ }^{25} \mathrm{C}_{4}$
$\Rightarrow{ }^{25} \mathrm{C}_{4}=\frac{25 !}{(25-4) ! \times 4 !}$
$\Rightarrow{ }^{25} \mathrm{C}_{4}=\frac{25 !}{21 ! \times 4 !}$
$\Rightarrow{ }^{25} \mathrm{C}_{4}=\frac{25 \times 24 \times 23 \times 22 \times 21 !}{21 ! \times 4 !}$
$\Rightarrow{ }^{25} \mathrm{C}_{4}=\frac{25 \times 24 \times 23 \times 22}{4 \times 3 \times 2 \times 1}$
$\Rightarrow{ }^{25} \mathrm{C}_{4}=12650$ possible ways.
Ans: In 12650 ways, from a class of 25 students, 4 can be chosen for a competition.
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