The English alphabet consists of 5 vowels (a, e, i, o, u) and 21 consonants. Vow

Question

The English alphabet consists of 5 vowels (a, e, i, o, u) and 21 consonants. Vowels are written on blue colored beads, and consonants are written on green colored beads. In the following, the size of a subset refers to the number of items in the subset (i.e the cardinality) A. The number of ways of choosing two nonempty subsets, with one subset containing only blue beads, and the other containing only green beads. B. The number of ways of choosing two subsets, with one subset containing the bead with the vowel 'a', and the other subset containing the bead with the consonant b'

Explanation / Answer

a)

number of choosing non-empty subset from a set of size n = 2^n -1

note that 1 is subtracted from 2^n

as the case when all are not selected ,

and total number of ways = 2^n

hence

required answer = (2^5 -1)* ( 2^21 -1)

b)

note that a and b are already in two different subsets

number of beads left = 26-2 = 24

now each bead has 3 option

1) go to first subset (with vowel a)

2) got ot second subset (with b)

3) don't go in any subset

hence

required number of ways = 3^24

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