# 5.(a)  Out of  $\small 100$  students, two sections of $\small 40$ and $\small 60$ are formed. If you and your friend are among the $\small 100$ students, what is the probability that(a) you both enter the same section?

H Harsh Kankaria

Total number of students = 100

Let A and B be the two sections consisting of 40 and 60 students respectively.

Number of ways of selecting 2 students out of 100 students.= $^{100}\textrm{C}_{n}$

(a)

If both are in section A:

Number of ways of selecting 40 students out of 100 = $\dpi{100} ^{100}\textrm{C}_{40}$  (The remaining 60 will automatically be in section B!)

Remaining 38 students are to be chosen out of (100-2 =) 98 students

$\dpi{100} \therefore$ Required probability if they both are in section A = $\frac{^{98}\textrm{C}_{38}}{^{100}\textrm{C}_{40}}$

Similarly,

If both are in section B:

Number of ways of selecting 60 students out of 100 = $\dpi{100} = ^{100}\textrm{C}_{60} = ^{100}\textrm{C}_{40}$  (The remaining 40 will automatically be in section A!)

Remaining 58 students are to be chosen out of (100-2 =) 98 students

$\dpi{100} \therefore$  Required probability if they both are in section B = $\frac{^{98}\textrm{C}_{58}}{^{100}\textrm{C}_{60}}$

Required probability that both are in same section = Probability that both are in section A + Probability that both are in section B

= $\frac{^{98}\textrm{C}_{38}}{^{100}\textrm{C}_{40}}+\frac{^{98}\textrm{C}_{58}}{^{100}\textrm{C}_{60}}$

$\\ = \frac{^{98}\textrm{C}_{38}+^{98}\textrm{C}_{58}}{^{100}\textrm{C}_{40}} \\ \\ =\frac{\frac{98!}{38!.60!} + \frac{98!}{58!.40!}}{\frac{100!}{40!.60!}}$

$=\frac{85}{165} = \frac{17}{33}$

Hence, the required probability that both are in same section is $\dpi{100} \frac{17}{33}$

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