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Corretore
Corretore asked in Science & MathematicsMathematics · 4 years ago

Combination Question?

In how many ways can letters of the word BEHAVIOUR be arranged so that at most 2 vowels are together?

I considered 2 cases.

First case: All vowels are separated. I got 2880 ways.

Second case: 2 vowels are together. I cannot do it.

Please give me a clear explanation.

Thanks.

1 Answer

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  • ?
    Lv 7
    4 years ago

    V = vowel

    C = consonant

    1st case:

    All vowels are separated: V C V C V C V C V

    Only 1 such arrangement of V's and C's

    5! ways to arrange the vowels

    4! ways to arrange the consonants

    5! * 4! = 2880

    So that's ok

    2nd case:

    Think of the 2 vowels that are together as a single vowel.

    So there are 4 "vowels" and 4 consonants, with no 2 "vowels" side by side:

    V C V C V C V C

    V C V C V C C V

    V C V C C V C V

    V C C V C V C V

    C V C V C V C V

    So there are 6 arrangements of V's and C's

    There are 5*4 = 20 ways to pick the 2 vowels that are together

    There are 4! ways to arrange the 4 "vowels"

    There are 4! ways to arrange the 4 consonants

    5 * 20 * 4! * 4! = 57600

    Total number of arrangements = 2880 + 57600 = 60400

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