First slide

Theorems of probability

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 x₃= 30 is

Moderate

A

$\frac{^{20} C_{2}}{^{50} C_{5}}$
B

$\frac{^{2} C_{2}}{^{50} C_{5}}$
C

$\frac{^{20} C_{2} \times^{29} C_{2}}{^{50} C_{5}}$
D

None of these

Solution

$Five tickets out of 50 {can drawn in}^{50} C_{5} ways. Since x_{1} < x_{2} < x_{2} < x_{4} < x_{5} and x_{2} = 30, x_{1}, x_{2} < 30, i.e. x_{1} and x_{2} should come$
$from tickets numbered 1 and 29 {and this may happen in}^{29} C_{2} ways$
$Remaining ways, i.e. x_{4}, x_{5} > 30, should come from$
$20 tickets numbered 31 to 50 {in}^{20} C_{2} ways.$
$So, favourable number of cases =^{29} {C_{2}}^{29} C_{2}$
$Hence, required probability = \frac{^{29} C_{2} \times^{29} C_{2}}{^{50} C_{5}}$

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