But the reality is that roughly half of the 5,000 American troops now in Iraq are likely to be involved in the operation, which could eventually require 30,000 Iraqi and Kurdish troops.

About 200 to 300 of the Americans are Special Operations commandos advising Iraqi and Kurdish troops — a mission that will put them a few miles behind the front lines, communicating with Iraqi soldiers via radio. A few dozen are forward air controllers, who are already calling in airstrikes against Islamic State targets, Lt. Gen. Stephen Townsend, the American commander in charge of the coalition in Iraq and Syria, said on Monday.