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Math Help - Refreshment of permutation and combination

  1. #1
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    Refreshment of permutation and combination

    (1) How many four letter words can be formed using the letters of the word EQUATION so that the letter Q is included in each word ?

    isn' t the answer 7x6x5 = 210 ... Am i correct ?


    (2) How many five letter words can be formed by picking 3 vowels and 2 consonants from the word REACTION such that two consonants always remain together ?
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  2. #2
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    Quote Originally Posted by mathaddict View Post
    (1) How many four letter words can be formed using the letters of the word EQUATION so that the letter Q is included in each word ?

    isn' t the answer 7x6x5 = 210 ... Am i correct ?
    No

    If Q is at the first place then you have 7 choices for the second letter then 6 choices for the third one and finally 5 choices for the fourth one
    That makes 7x6x5 = 210
    But if Q is at the second place you have also 210 possibilities. The same if Q is at the third place and the same for the fourth place.
    Therefore there are 210 x 4 = 840 words

    Another way is to calculate the number of group of 4 letters containing Q
    There are \binom{7}{3} = \frac{7!}{3!\: 4!} = 35 groups of 4 letters with Q (the same number as the number of groups of 3 letters among 7 letters)
    For each group of 4 letters you can build 4! = 24 words
    The total number of words is 35 x 24 = 840
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  3. #3
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    Hello, mathaddict!

    Another approach . . .


    (1) How many 4-letter words can be formed using the letters of the word EQUATION
    so that the letter Q is included in each word?
    The letter Q will be included.
    Select 3 letters from the other 7 letters: . {7\choose 3} \,=\,{\color{blue}35} ways.

    We have a set of 4 letters; they can be arranged in: 4! \,=\,{\color{blue}24} ways.

    Therefore, there are: . 35 \times 24 \:=\:\boxed{{\color{blue}840}} possible words.



    (2) How many 5-letter words can be formed by picking 3 vowels and 2 consonants
    from the word REACTION such that two consonants always remain together?
    There are 4 vowels and 4 consonants.

    Select 3 of the 4 vowels: . {4\choose3} \,=\,4 ways.
    Select 2 of the 4 consonants: . {4\choose2} \,=\,6 ways.

    . . We have: . 4 \times 6 \:=\:{\color{blue}24} selections of letters.


    We have 3 vowels and 2 consonants: . \{V,V,V,C,C\}

    Duct-tape the two consonants together: . \left\{V,V,V,\boxed{CC}\right\}

    These four "letters" can be arranged in: 4! \,=\,24 ways.
    But the two constants could be ordered either C_1C_2 or C_2C_1.
    . . Hence, there are: . 2 \times 24 \:=\:{\color{blue}48} orders.

    Therefore, there are: . 24 \times 48 \:=\:\boxed{{\color{blue}1152}} possible words.

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  4. #4
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    Quote Originally Posted by Soroban View Post
    Hello, mathaddict!

    Another approach . . .


    The letter Q will be included.
    Select 3 letters from the other 7 letters: . {7\choose 3} \,=\,{\color{blue}35} ways.

    We have a set of 4 letters; they can be arranged in: 4! \,=\,{\color{blue}24} ways.

    Therefore, there are: . 35 \times 24 \:=\:\boxed{{\color{blue}840}} possible words.

    Not so new !
    See my post
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