Find the number of ways the letters A,E,I,O,U,B,C,D,F,G can be arranged so that at least 4 vowels are together

Question

Find the number of ways the letters A,E,I,O,U,B,C,D,F,G can be arranged so that at least 4 vowels are together

Please assist in providing a step by step answer to help my daughter to understand the method to solve these kind of problems

Answer 1

Hint:

Break into two cases. The case where all five of the vowels are together, and the case where only exactly four of the vowels are together and the fifth vowel is elsewhere.

Arrange the consonants first leaving some room to either side, arrange the vowels separately, and then pick which space is used by the vowel cluster (and which space is used by the singleton vowel in the second case)

$(5!)(5!)\cdot 6 + (5!)(5!)\cdot 6\cdot 5$