Question

Out of 100 students, two sections of 40 and 60 are formed. If you and your friend are among the 100 students, what is the probability that.
(a) You both enter the same section?
(b) You both enter the different sections?

Answer 1

Total favourable outcomes in which both the students are in same section of 40 students ${=}^{98}{C}_{38}$
Also total favourable outcomes in which both the students are in same section of 60 students ${=}^{98}{C}_{58}$
(a) P(Both the students are in same section)
$=\frac{{}^{98}{C}_{38}{+}^{98}{C}_{58}}{{}^{100}{C}_{40}}=\frac{\frac{98!}{38!60!}+\frac{98!}{58!40!}}{\frac{100!}{40!60!}}=\frac{98![\frac{1}{38!60!}+\frac{1}{58!40!}]}{\frac{100!}{40!60!}}=\frac{\frac{98!\times 58!\times 38!}{38!\times 60!\times 58!\times 40!}(40\times 39+60\times 59)}{\frac{100}{40!60!}}=\frac{98!}{40!\times 60!}\times 5100\times \frac{40!\times 60!}{100!}=\frac{5100}{100\times 99}=\frac{17}{33}\text{(b) (Both students are in different section)}=1-\frac{17}{33}=\frac{16}{33}$