The number of subsets of a set is given by the formula: \[ \text{Number of subsets} = 2^n \] where \(n\) is the number of elements in the set. In this case, the set \(P = \{a, e, i, o, u\}\) contains 5 elements. Therefore, the number of subsets is: \[ 2^5 = 32 \] Thus, the set \(P\) will have 32 subsets.
The correct option is (D): \(32\)