Take an ordering of the consonants (there are 21! such orderings), for example:
bcdfghjklmnpqrstvwxyz
Enumerate the 22 "slots" between these consonants:
_b_c_d_f_g_h_j_k_l_m_n_p_q_r_s_t_v_w_x_y_z_
We can choose (respecting order) ₂₂P₅ lists of five slots to put the vowels aeiou in order.
The answer is therefore ₂₂P₅ ₓ 21!
