Question

In a lottery, person choses six different natural numbers at random from 1 to 20, and 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? [Hint: order of the numbers is not important.]

Answer 1

Total number of ways in which one can choose six different numbers from 1 to 20 = $C_6^{20}=⌊\frac{20}{⌊6⌊20-6}=\frac{⌊20}{⌊6⌊14}$
$=\frac{20x19x18x17x16x15}{1.2.3.4.5.6}=38760$
Hence, there are 38760 combinations of 6 numbers.
Out of these combinations, one combination is already fixed by the lottery committee.
∴ Required probability of winning the prize in the game =$\frac{1}{38760}.$