= Solution
Every seven-element subset of the hundred individually identifiable tiles is equally likely. Choose three of the thirty-four vowels and four of the sixty-four consonants; the favourable draw has no blank tiles. Thus
$$
\boxed{P=\frac{\binom{34}{3}\binom{64}{4}}{\binom{100}{7}}}.
$$
This is a <multivariate hypergeometric distribution> count, including the implicit factor $\binom20=1$ for blanks. Dividing by $\binom{98}{7}$ instead would incorrectly condition on having drawn no blanks.
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