An expression is formed using the numbers 7, 16, 25, and 27 according to the following rules. $\bullet$ Each of the four numbers is used exactly once. $\bullet$ The four numbers may be used in any order. $\bullet$ Exactly three operations are used; each one is either $+$ or $\times$. $\bullet$ An unlimited number of parentheses may be used. No two distinct expressions have the same simplified value. The two expressions below are not distinct, and therefore must be counted as only one value. What is the greatest number of distinct values, including the one below, that can be obtained when building expressions following these rules? $(7+16+27) \times 25 = 25 \times (27+7+16)$

Williamjwu8 Jul 27, 2020