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?

Answer 1

All numbers are different (given in question), this will be the same as picking r different objects from n objects which is ⁿc_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 ²⁰C₆ = \(\frac{(20)!}{(20-6)!\times(6)!}\)as ⁿC_r = \(\frac{(n)!}{(n-r)!\times(r)!}\)  i.e. ²⁰C₆ = 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

 = \(\frac{Total\,no.of\,Desired\,outcomes}{Total\,no.of\,outcomes}\)

Therefore, the probability of winning a prize is 

= \(\frac{1}{38760}\)

Conclusion: Probability of winning the prize in the game is \(\frac{1}{38760}\)