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Seven ducks, an adult male, an adult female, and 5 ducklings (one of

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Bunuel wrote:
Seven ducks, an adult male, an adult female, and 5 ducklings (one of whom is particularly unattractive), are walking in a straight line from the barn to the pond. The line is led by and ended with an adult duck and the ugly duckling must be in the exact center of the line. How many different ways can this line be formed?

A. 24
B. 48
C. 60
D. 120
E. 240

We have 7 total ducks, including one adult male, one adult female, and one ugly duckling. We need to determine the number of ways the ducks can line up with the adult ducks at each end and the ugly duckling in the center.

In the first spot, we have 2 choices (since it can be 1 of the 2 adult ducks).

In the second spot, we have 4 choices (since it can be any 1 of the 4 non-ugly ducklings).

In the third spot, we have 3 choices (since it can be any 1 of the 3 remaining non-ugly ducklings).

In the fourth spot, i.e., the center spot, we have only 1 choice (since it must be the ugly duckling).

In the fifth spot, we have 2 choices (since it can be any 1 of the 2 remaining non-ugly ducklings).

In the six spot, we have 1 choice (since it must be the last remaining non-ugly duckling).

In the seventh spot, i.e., the last position, we have 1 choice (since it must be the other adult duck).

Thus, the number of ways the ducks can lineup with an adult duck at each end and the ugly duckling in the middle is 2 x 4 x 3 x 1 x 2 x 1 x 1 = 48.

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we can arrange an ugly duck in one way , like wise from 2 adults ducks one of which can be arranged in 2 ways and the other one can be in 1 ways , rest can be arranged in 4!
hence B
2x4x3x2x1x1x1= 48
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