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Virginia writes down the value of the $$100$$ products $$a \times b$$, where $$a$$ and $$b$$ represent integers from $$0$$ to $$9$$, inclusive. How many distinct numbers does Virginia write?

Oct 15, 2023

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Virginia writes down the value of 100 products of integers from 0 to 9. For each product, she has 10 choices for the first integer and 10 choices for the second integer, for a total of 10⋅10=100 possibilities. However, some of these products will be the same.

For example, both 2⋅5 and 5⋅2 are equal to 10, so these two products are counted twice. To avoid this overcounting, we can use the following formula to count the number of distinct products:

(rn+r−1​)

where n is the number of distinct elements in the set and r is the number of elements in each product. In this case, we have n=10 distinct elements (the digits 0 to 9) and r=2 elements in each product. So, the number of distinct products is:

(210+2−1​)=(211​)

Using the combination formula, we can calculate that there are (211​)=55 distinct products. Therefore, Virginia writes down 55​ distinct numbers.

Oct 15, 2023