In Which of the following colonial wars did the American colonists capture the fort at Louisbourg and begin to crave domination over the entire western region that included the Ohio Valley?

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To determine which colonial war involved the capture of the fort at Louisbourg by American colonists and their desire for domination over the western region, including the Ohio Valley, we need to examine the context and historical events.

The options to consider are:

1. French and Indian War (1754-1763)
2. King Philip's War (1675–1676)
3. Bacon's Rebellion (1676)
4. Pontiac's Rebellion (1763-1766)

To find the correct answer, we can eliminate choices by assessing the given events and their historical significance.

- King Philip's War (1675–1676) primarily occurred between English colonists and Native American tribes in New England. It does not match the criteria of American colonists capturing the fort at Louisbourg or their desire for domination over the Ohio Valley.

- Bacon's Rebellion (1676) was an uprising in colonial Virginia against the rule of Governor William Berkeley. It focused on political and economic grievances, and it does not involve capturing the fort at Louisbourg or ambition for western domination.

- Pontiac's Rebellion (1763-1766) was an uprising by various Native American tribes against British rule in the Great Lakes region after the French and Indian War. Although it followed the time frame of the capture of Louisbourg, it does not involve American colonists capturing the fort or aspiring to control the Ohio Valley.

This leaves us with the French and Indian War (1754-1763), which was a conflict between the British and French colonial powers fought primarily in North America. American colonists did capture the fort at Louisbourg in 1745 during British attempts to gain control of Acadia. The American colonists' desire for domination over the Ohio Valley is significant because it was one of the major disputed regions between the two powers, leading to the outbreak of conflict.

Therefore, the correct answer is the French and Indian War (1754-1763).