Question

A bag contains \[50\] tickets numbered\[1,2,3,.....,50\] of which five are drawn at random and arranged in ascending order of magnitude \[({x_1} < {x_2} < {x_3} < {x_4} < {x_5})\]. The probability that \[{x_3} = 30\] is
A. \[\dfrac{{{}^{20}{C_2}}}{{{}^{50}{C_5}}}\]
B. \[\dfrac{{{}^{29}{C_2}}}{{{}^{50}{C_5}}}\]
C. \[\dfrac{{{}^{29}{C_2} \times {}^{20}{C_2}}}{{{}^{50}{C_5}}}\]
D. None of these

Answer 1

Hint:We are given that number \[30\] is on \[{3^{rd}}\] position. So, we have to choose two numbers from \[1\] to \[29\] numbers and choose two numbers from \[31\] to \[50\]
Then select cards \[5\] from \[1\] to \[50\] and find the required probability.

Complete step-by-step solution:
We have been given that a bag contains \[50\] tickets numbered \[1,2,3,.....,50\] of which five are drawn at random and arranged in ascending order of magnitude \[({x_1} < {x_2} < {x_3} < {x_4} < {x_5})\].
Now we assume that E= Event of getting \[5\] tickets in ascending order. Since it is given that number \[30\] is on \[{3^{rd}}\] position. So, we have to choose two numbers from \[1\] to \[29\] numbers and choose two numbers from \[31\] to \[50\].
Therefore, the number of ways of selecting \[5\] tickets out of \[50\] = \[{}^{50}{C_5}\] ways.
Now it is given that \[{x_1} < {x_2} < {x_3} < {x_4} < {x_5}\] and \[{x_3} = 30\]
\[{x_1},{x_2} <30\].
That is \[{x_1}\] and \[{x_2}\] should come from tickets numbered \[1\] and \[29\].
Thus, the number of ways of selecting \[{x_1}\] and \[{x_2} = {}^{29}{C_2}\] ways.
Now the remaining \[{x_4},{x_5} > 30\] should come from \[20\] tickets numbered \[31\] to \[50\].
Thus, the number of ways of selecting \[{x_4},{x_5} = {}^{20}{C_2}\] ways.
So, the favourable number of cases are \[ = {}^{29}{C_2} \times {}^{20}{C_2}\]
Hence, the required probability is \[ = \dfrac{{{}^{29}{C_2} \times {}^{20}{C_2}}}{{{}^{50}{C_5}}}\]

Hence option(C) is correct

Note:Since we are picking \[5\] distinct numbers, there is only one way of arranging them in increasing order so we do not have to worry about the arrangement. We just need to pick the right
numbers and find the required probability.