Question

In a lottery, four random numbers from $1-20$ are chosen. A winning ticket matches the four numbers in any order. What is the probability of a winning ticket?

• A. $\frac{1}{4845}$
• B. $\frac{2}{4845}$
• C. $\frac{3}{4845}$
• D. $\frac{4}{4845}$

Answer 1

The correct option is A $\frac{1}{4845}$

Step 1: Combination of selecting four numbers is ${}^{20}{\text{C}}_4$

$\Rightarrow {}^{20}{\text{C}}_4=\frac{20\times 19\times 18\times 17}{4\times 3\times 2}$
$=19\times 17\times 5\times 3$
$=4845$

Step 2: Combination of single winning ticket is ${}^4{\text{C}}_4$
$\Rightarrow {}^4{\text{C}}_4=1$

Step 3: Probability of winning ticket
$=\frac{{}^4{\text{C}}_4}{{}^{20}{\text{C}}_4}$
$\Rightarrow \frac{{}^4{\text{C}}_4}{{}^{20}{\text{C}}_4}=\frac{1}{4845}$