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To determine if it is possible to include 10 children and 5 adults within the budget, we need to calculate the total cost for this scenario.
The cost per child is $10.50, and there are 10 children, so the cost for the children would be 10 * $10.50 = $<<10*10.5=105>>105.
The cost per adult is $5.50, and there are 5 adults, so the cost for the adults would be 5 * $5.50 = $<<5*5.5=27.5>>27.5.
Adding the cost for the children, adults, and the initial cost of $150, the total cost would be $105 + $27.5 + $150 = $<<105+27.5+150=282.5>>282.5.
The total cost of $282.5 is within the budget of $300, so it is possible to include 10 children and 5 adults while staying within the budget.