Four institutes, A, B, C, and D, had contracts with four vendors W, X, Y, and Z during the ten calendar years from 2010 to 2019. The contracts were either multi-year contracts running for several consecutive years or single-year contracts. No institute had more than one contract with the same vendor. However, in a calendar year, an institute may have had contracts with multiple vendors, and a vendor may have had contracts with multiple institutes.
Over the ten years, each institute had exactly two contracts with two different vendors, and each vendor had exactly two contracts with two different institutes.
The following facts are known about the contracts:
1. Vendor Z had at least one contract in every year.
2. Vendor X had one or more contracts in every year up to 2015, but no contract in any year after that.
3. Vendor Y had contracts in 2010 and 2019. Vendor W had contracts only in 2012.
4. There were five contracts in 2012.
5. There were exactly four multi-year contracts: Institute B had a 7-year contract, Institute D had a 4-year contract, and Institutes A and C had one 3-year contract each. The remaining four contracts were single-year contracts.
6. Institute C had one or more contracts in 2012 but did not have any contract in 2011.
7. Institutes B and D each had exactly one contract in 2012. Institute D did not have any contract in 2010.
Which institutes had multiple contracts during the same year?
Correct Answer :
A and B only
Solution :
Let's look at the contract overlaps for each institute over the decade:
- Institute A: Had a 3-year contract with Vendor X (2010–2012) and a single-year contract with Vendor W in 2012. Thus, in 2012, Institute A had contracts with both X and W (multiple contracts in the same year).
- Institute B: Had a 7-year contract with Vendor Z (2010–2016) and a single-year contract with Vendor Y in 2013. Thus, in 2013, Institute B had contracts with both Z and Y (multiple contracts in the same year).
- Institute C: Had a single-year contract with Vendor Y in 2010, a single-year contract with Vendor W in 2012, and a 3-year contract with Vendor Z (2017–2019). There is no year in which these contracts overlap.
- Institute D: Had a 4-year contract with Vendor X (2012–2015) and a single-year contract with Vendor Y in 2019. These do not overlap.
Therefore, only Institutes A and B had multiple contracts during the same year.
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