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I have a circuit that has 4 inputs and am only allowed to use 5-, 3- and 2-input NAND gates. Yes this is an assignment question but I have already solved the design of the circuit.

I just don't know how to use (if possible) 4 outputs with a 5-input NAND gate.

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You can use 5 input nand and only use 4 of the inputs. Make sure you tie the unused input high (logic 1).

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    \$\begingroup\$ Another solution is to fan out one of the inputs into two inputs of the NAND gate (similarly to how a NOT is made from a binary NAND). not(P and P and Q) is not(P and Q). You solution relies on P and T = P, and this one on P and P = P. \$\endgroup\$ – Kaz Apr 6 '13 at 3:39

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