+30 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?
+721 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.
