The 8 leaders of the G8 nations convene in Rome and stand in a row as they get ready to have some pictures of them taken by the press. What is the probability that the picture that the New York Times' editors will randomly select for publishing the next day is one in which Berlusconi is not standing next to Obama? (assuming that there are pictures of all possible standing arrangements).
A. 1/8
B. 1/4
C. 1/2
D. 3/4
E. 7/8
P(good outcome) = 1 - P(bad outcome).
Total arrangements:
Number of ways to arrange the 8 leaders = 8!.
Bad arrangements:
In a bad arrangement, B and O stand adjacent to each other.
Let [BO] represent B and O standing adjacent to each other and R, S, T, U, V, and W represent the other 6 leaders.
Number of ways to arrange the 7 elements [BO], R, S, T, U, V, and W = 7!.
Since BO can be reversed to OB, we multiply by 2:
2*7!.
Thus:
P(bad outcome) = (bad arrangements)/(total arrangements) = (2*7!)/8! = 1/4.
Thus:
P(good outcome) = 1 - 1/4 = 3/4.
The correct answer is
D.
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