TCS Company Numerical Ability Permutation and Combination

Q. 10 people meet and shake hands. The maximum number of handshakes possible if there is to be no "cycle" of handshakes is (A cycle of handshakes is a sequence of k people a1, a2, ......, ak (k > 2) such that the pairs {a1, a2}, {a2, a3}, ......, {ak-1, ak}, {ak, a1} shake hands).
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TCS Other Question

Q. 36 people {a1, a2, ..., a36} meet and shake hands in a circular fashion. In other words, there are totally 36 handshakes involving the pairs, {a1, a2}, {a2, a3}, ..., {a35, a36}, {a36, a1}. Then size of the smallest set of people such that the rest have shaken hands with at least one person in the set is
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there are 40 barreals are there and one is poisoned. If a drop is consumed he will die in 14 hrs then how many least mice are required to find the poisoned barreal?if we have only 24 hour.