| Out of 30 consecutive positive integers, two are chosen at random. Find the probability of their sum to be even. |
A) \[\frac{14}{29}\]
B) \[\frac{11}{29}\]
C) \[\frac{17}{29}\]
D) \[\frac{19}{29}\]
Correct Answer: A
Solution :
(a) The total number of ways of choosing two out of 30, i.e. \[{}^{30}{{C}_{2}}\]ways. Let E = Event of choosing two numbers such that their sum is even. \[\therefore \]\[P\,\,(E)=\frac{n\,\,(E)}{n\,\,(S)}=\frac{{}^{15}{{C}_{2}}+{}^{15}{{C}_{2}}}{{}^{30}{{C}_{12}}}\] \[=\frac{105+105}{435}=\frac{210}{435}=\frac{14}{29}\]
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