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let P,Q,R as a lock and to open lock at least two keys are required

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As atleast two people must enter keys into assigned lock,so the condition for vault to open is:-

k1k2 + k2k3 + k1k3

3 nand gates for there complement

(k1k2)' and 2 others.

((k1k2)'.(k2k3)')' 4th gate

(4th gate.(k1k3)')' 5 th gate

(4thgate.5thgate)' 6th Nand gate which gives k1k2+k2k3+k1k3

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