What is (a) the highest, (b) the lowest total resistance that can be secured by combinations of four coils of resistance 4 ohm, 8 ohm, 12 ohm, 24 ohm?

The total resistance of resistors or coils in series is the sum of their individual resistances, and for resistors or coils in parallel, the reciprocal of the total resistance is the sum of the reciprocals of their individual resistances.

(a) Highest Total Resistance:
For the highest total resistance, you would connect the resistors in series because the total resistance in a series connection is the sum of the individual resistances.

R_total, series = R₁ + R₂ + R₃ + R₄

R_total, series = 4 + 8 + 12 + 24 = 48Ω
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So, the highest total resistance is 48Ω when the resistors are connected in series.

(b) Lowest Total Resistance:
For the lowest total resistance, you would connect the resistors in parallel because the total resistance in a parallel connection is given by the reciprocal of the sum of the reciprocals of the individual resistances.

1/R_total, paraller = 1/R₁ + 1/R₂ + 1/R₃ + 1​/R₄

1/R_total, paraller = 6/24 + 3/24 + 2/24 + 1​/24

1/R_total, parallel = 12/24
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total, parallel = 2Ω
Rtotal, parallel = 2Ω