Two brothers and three sisters form a single line for a photograph. The two boys refuse to stand next to each other.
How many different line-ups are possible?
Let the five positions in the photograph be numbered 1, 2, 3, 4, 5. Then the boys may occupy a total of six positions: 1 and 3; 1 and 4; 1 and 5; 2 and 4; 2 and 5; 3and 5. For each of these positions, the boys may be arranged in two ways as they can interchange places. So there are 12 ways of positioning the boys.
For each of these, the girls must be placed in three positions. In each case, the first girl may choose any one of three positions, the second girl may choose either of two positions and then there is just one place remaining for the third girl. So for each arrangement of the two boys there are 3 × 2 × 1 different ways of arranging the three girls.
Therefore the total number of line-ups is 12 × 6 = 72.
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