A probability problem by Shenal Kotuwewatta

We can either have $4$ vowels and $3$ consonants (case 1), or $3$ vowels and $4$ consonants (case 2).

Case 1: the probability is $\left(\dfrac{5}{26}\right)^4\left(\dfrac{21}{26}\right)^3=\dfrac{5788125}{8031810176}$

Case 2: the probability is $\left(\dfrac{5}{26}\right)^3\left(\dfrac{21}{26}\right)^4=\dfrac{24310125}{8031810176}$

Adding these two probabilities together, we get that final probability is $\dfrac{1157625}{308915776}$ , so our final answer is $\boxed{401}$ .

There are two cases the word is pronounceable:

1> It has 4 vowels (V) and 3 consonants (C), so we have $5^{4}*21^{3}$ ways to arrange them in this order: V-C-V-C-V-C-V

2> It has 4 C and 3 V, and there are $5^{3}*21^{4}$ ways to arrange it in this order: C-V-C-V-C-V-C

The probability of this word is $\frac{5^{4}*21^{3}+5^{3}*21^{4}}{26^{7}}=\frac{5*5^{3}*21^{3}+21*5^{3}*21^{3}}{26^{7}}=\frac{26*5^{3}*21^{3}}{26^{7}}=\frac{5^{3}*21^{3}}{26^{6}}=\frac{1157625}{308915776}$

the numerator and denominator at the most right hand equation are coprime and their sum is 310073401. The last 3 digits is 401.