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.

In how many years during this period was there only one contract?

To determine the number of contracts in each year, we can reconstruct the schedule of contracts for the four institutes (A, B, C, D) and four vendors (W, X, Y, Z) over the ten-year period (2010 to 2019) using the given clues:

1. Multi-year Contracts: There are exactly four multi-year contracts: B (7 years), D (4 years), A (3 years), and C (3 years). The other four are single-year contracts.

2. Vendor Z: Z is active in all 10 years. Since Z can have at most two contracts (which must be with different institutes), these two contracts must cover the entire 10-year period. The only way to cover 10 years with two contracts is if one is B's 7-year contract (2010–2016) and the other is C's 3-year contract (2017–2019).

3. Vendor X: X is active from 2010 to 2015. This 6-year period must be covered by X's two contracts. These must be A's 3-year contract (2010–2012) and D's 4-year contract (2012–2015).

4. Single-Year Contracts:
- W is active only in 2012 (contracts with A and C).
- Y is active in 2010 (contract with C) and 2019 (contract with D).

Now, we can count the number of active contracts in each year:

- 2010: 3 contracts (A-X, B-Z, C-Y)
- 2011: 2 contracts (A-X, B-Z)
- 2012: 5 contracts (A-X, B-Z, D-X, A-W, C-W)
- 2013: 3 contracts (B-Z, D-X, B-Y)
- 2014: 2 contracts (B-Z, D-X)
- 2015: 2 contracts (B-Z, D-X)
- 2016: 1 contract (B-Z)
- 2017: 1 contract (C-Z)
- 2018: 1 contract (C-Z)
- 2019: 2 contracts (C-Z, D-Y)

From this schedule, there was only one contract in the years 2016, 2017, and 2018. Thus, there are exactly 3 years with only one contract.