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# Digit problem

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How many 3-digit positive integers can be added to $$463$$ such that no carry occurs when adding the digits?

Jan 31, 2022

#1
+1374
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A carry occurs when the place values have a sum $$\geq10$$

For the $$1$$'s digit, there are $$7$$ choices $$(0,1,2,3,4,5,6)$$

For the $$10$$'s digit, there are $$4$$ choices $$(0,1,2,3)$$

For the $$100$$'s digit, there are $$5$$ choices $$(1,2,3,4,5)$$. Remember, we can't use $$0$$ for the hundred's digit, because then the number wouldn't be 3-digits.

This means that there are $$7 \times 4 \times 5$$ numbers. Thus, the answer is $$\color{brown}\boxed {140}$$

Jan 31, 2022