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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.]
$\frac{1}{38760}$
$\frac{1}{38760}$
$\frac{1}{38760}$
$\frac{1}{38760}$
Solution
Total number of ways in which one can choose six different numbers from $1$ to $2.$
$=^{20} C_{6}=\frac{\lfloor {20}}{\lfloor {6\lfloor {20-6}}}=\frac{\lfloor {20}}{\lfloor {6\lfloor {14}}}$
$=\frac{20 \times 19 \times 18 \times 17 \times 16 \times 15}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6}$ $=38760$
Hence, there are $38760$ combinations of $6$ numbers.
Out of these combinations, one combination is already fixed by the lottery committee.
$\therefore$ Required probability of winning the prize in the game $=\frac{1}{38760}$
