Solution: The possible distributions of cards are
The probability of a full house is the sum of these three numbers, divided by C(52,7). The probability is approximately .02596
Solution: We first find the number of ways to choose slots for the vowels. There are 9 vowels and 11 consonants. Represent the vowel slots by bars and the consonant slots by stars. Each pair of bars must have at least one star between them. There are 8 gaps between bars, so the number of such arrangements is equal to the number of ways to arrange 11-8=3 stars among the bars with no restriction on where they must go. The number of possible arrangements of 3 stars and 9 bars is C(12,3). Thus, there are C(12,3) ways to choose slots for the vowels.
The vowels consist of 3 I's, 2 E's, 2 A's, and 2 O's, so the number of ways to arrange the vowels in their chosen slots is
The consonants consist of 2 M's, 4 N's, 1 P, 1 L, 2 S's, and 1 T, so the number of ways to arrange the consonants in their chosen slots is
Thus, the total number of possible arrangements is