Question: In which of the following years were there two or more contracts?
Explanation:
Given:
It is known that over the decade, the institutes each got into two contracts with two of these vendors, and each vendor got into two contracts with two of these institutes.
Therefore, considering point (V), A had one 3-year contract and one single-year contract. B had one 7- year contract and one single-year contract. C had one 3-year contract and one single-year contract. D had one 4-year contract and one single-year contract.
Using point (III), we can conclude that Y and W have only one-year contracts. Thus, 4 one-year contracts were with Y and W.
Using points (III) and (IV), W had two single-year contracts in 2012. Y did not have any contract in 2012. So, among Z and X, either Z has 2 contracts or X has two contract in 2012.
Using point (II) and (III), X, Y and W did not have 7-year contract. Therefore using point (I) we can conclude that Z must have had 7-year contract with Institute B. Also, another contract Z has must be a 3-year or a 4-year contract.
Case (i): Z has a 4-year contract.
Then X will have two 3-year contracts. One from 2010-2012 and other from 2013-2015.
So Z has to have two contracts in 2012. The 4-year contract must have been from 2010-2013 and the 7- year contract from 2013-2019 OR the 7-year contract must have been from 2010-2016 and the 4-year contract from 2016-2019
But in that case, there will be only 4 contracts in 2012. Therefore this case is invalid.
Thus, Z had a 3-year contract either with institute A or C. Therefore, X had one 4-year with D and one 3- year contract. As D did not have contract in 2010 and C did not have contract in 2011, A had contract in 2010-2012 with Vendor X and D had contract from 2012-2015 with vendor X. Again as institute C did not have a contact in 2011, B had a 7-year contract (2010-2016) with vendor Z and C had a 3-year contract (2017-2019) with vendor Z.
As institutes B and D each had exactly one contract in 2012, Remaining contracts must be with A and C.
Therefore, both contracts of W must be single-year contracts with institutes A and C. As D did not have any contract in 2010, Y must have had a single-year contract with B in 2010 and with D in 2019.
Thus, we have
Among the given options, there were two or more contracts in 2015.
Hence, option (c).