# Q : 11     In a lottery, a 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.]

Total numbers of numbers in the draw = 20

Numbers to be selected = 6

$\dpi{100} \therefore$ n(S) = $\dpi{100} ^{20}\textrm{C}_{6}$

Let E be the event that six numbers match with the six numbers fixed by the lottery committee.

n(E) = 1 (Since only one prize to be won.)

$\dpi{100} \therefore$ Probability of winning =

$\dpi{100} P(E) = \frac{n(E)}{n(S)}$$\dpi{100} = \frac{1}{^{20}\textrm{C}_{6}} = \frac{6!14!}{20!}$

$\dpi{100} = \frac{6.5.4.3.2.1.14!}{20.19.18.17.16.15.14!}$

$\dpi{100} = \frac{1}{38760}$

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