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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

Jay C
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1 Answers1

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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)

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

JMoravitz
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  • This makes sense – Jay C May 03 '18 at 04:49
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    This makes sense

    case 1 : All 5 vowels together = 6! *5! = 86400

    case 2 : Only 4 vowels together

    -- C -- C -- C -- C -- C -- 2 vowel groups can take only those position marked as --

    5! - ways to arrange consonants 5C4 - choosing 4 from 5 vowels 4! - arranging 4 vowels 6P2 - filling 6 gaps with 2 vowel groups = 5! * 5c4 * 4! * 6p2 = 432000

    Total = 86400+43200 = 518400

     – Jay C May 03 '18 at 04:56