To find the area that the goats will be allowed to roam in, we need to first calculate the area of the larger fenced area for the goats and then subtract the area of the smaller fenced garden.
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Calculate the area of the larger fenced area: \[ \text{Area of goats' fenced area} = \text{length} \times \text{width} = 12 , \text{yd} \times 15 , \text{yd} = 180 , \text{yd}^2 \]
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Calculate the area of the smaller garden: \[ \text{Area of garden} = \text{length} \times \text{width} = 3 , \text{yd} \times 6 , \text{yd} = 18 , \text{yd}^2 \]
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Subtract the area of the garden from the area of the goats' fenced area: \[ \text{Area for goats} = \text{Area of goats' fenced area} - \text{Area of garden} = 180 , \text{yd}^2 - 18 , \text{yd}^2 = 162 , \text{yd}^2 \]
Therefore, the area that the goats will be allowed to roam in is \(162 , \text{yd}^2\).
The correct response is: 162 yd².