# In a lottery ,a person choose six,different natural numbers at random from 1 to 20 and if these six numbers match with the six numbers already fixed by 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]

$\begin{array}{1 1}(A)\;\large\frac{1}{38760}\\(B)\;\large\frac{1}{28760}\\(C)\;\large\frac{1}{18760}\\(D)\;\text{None of these}\end{array}$

Toolbox:
• This problem can be solved by combination method (i.e) $nC_r=\large\frac{n!}{r!(n-r)!}$
Step 1:
Given :
6 numbers out of 20 numbers are chosen
$\therefore$ These numbers can be chosen in $20C_6$ ways and only one combination of the number is correct
Let the probability of winning the prize be E
$\therefore n(E)=1$
Only one prize can be won
Step 2:
$n(S)=20C_6=\large\frac{20!}{6! 14!}$
$\Rightarrow \large\frac{20\times 19\times 18\times 17\times 16\times 15}{6\times 5\times 4\times 3\times 2\times 1}$
$\Rightarrow 38760$
Step 3:
Probability =$\large\frac{n(E)}{n(S)}$
$\Rightarrow \large\frac{1}{38760}$
Hence (A) is the correct answer.