# In a lottery, a person chooses six different numbers at random from 1 to 20.

Question:

In a lottery, a person chooses six different numbers at random from 1 to 20. If these six numbers match with the six numbers already fixed by the lottery committee, he wins the prize. What is the probability of winning the prize in the game?

Solution:

All numbers are different (given in question), this will be the same as picking r different objects from n objects which is ${ }^{n} \mathrm{C}_{r}$

Here, n= 20 and r = 6(as we have to pick 6 different objects from 20 objects)

Now we shall calculate the value of ${ }^{20} \mathrm{C}_{6}=\frac{(20) !}{(20-6) ! \times(6) !}$ as ${ }^{\mathrm{n}} \mathrm{Cr}_{\mathrm{r}}=\frac{(\mathrm{n}) !}{(\mathrm{n}-\mathrm{r}) ! \times(\mathrm{r}) !}$

i.e. ${ }^{20} \mathrm{C}_{6}=38760$

Therefore, 38760 cases are possible, and in that only one them has prize, i.e. total no.of desired outcome is 1

As we know,

Probability of occurrence of an event

$=\frac{\text { Total no.of Desired outcomes }}{\text { Total no. of outcomes }}$

Therefore, the probability of winning a prize is

$=\frac{1}{38760}$

Conclusion: Probability of winning the prize in the game

is $\frac{1}{38760}$