Five numbers are taken out from the numbers 1-30 and arranged in ascending order. The probability that the third number will be 20 is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle\frac{^{19}C_{2}\times{}^{10}C_{2}}{^{30}C_{5}}\)
The number of ways to choose 5 numbers out of 30 is \(^{30}C_{5}\).
The third number in ascending order is 20 exactly when 20 is chosen, two numbers are smaller than 20 and two are larger:
\(\text{two from } 1, 2, \ldots, 19 \text{ (19 numbers)}:\ ^{19}C_{2}\)
\(\text{two from } 21, 22, \ldots, 30 \text{ (10 numbers)}:\ ^{10}C_{2}\)
So
\( \displaystyle P = \frac{^{19}C_{2}\times{}^{10}C_{2}}{^{30}C_{5}} \)