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Math Help - 5 Basic tastes, how many combinations?

  1. #1
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    5 Basic tastes, how many combinations?

    Question: If the papillae (bumps on your tongue) can sense 5 basic tastes and any combination of those tastes produce a unique taste, how many possible unique taste can you have?

    Misconception: 4+3+2+1 = 10 unique tastes, but that only accounts for non-repetitive pairs of tastes. In this problem, any number of the 5 basic, different taste can account for 1 unique taste (e.g., taste A,B,C can mix to create 1 unique taste). With that in mind, how many possible unique taste can you have?

    My 2nd answer: 16 unique tastes.
    (4+3) + (3+2) + (2+1) + 1

    Please do the question and see if your answer matches up with my 2nd answer.
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  2. #2
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    Quote Originally Posted by Masterthief1324 View Post
    Question: If the papillae (bumps on your tongue) can sense 5 basic tastes and any combination of those tastes produce a unique taste, how many possible unique taste can you have?

    Misconception: 4+3+2+1 = 10 unique tastes, but that only accounts for non-repetitive pairs of tastes. In this problem, any number of the 5 basic, different taste can account for 1 unique taste (e.g., taste A,B,C can mix to create 1 unique taste). With that in mind, how many possible unique taste can you have?

    My 2nd answer: 16 unique tastes.
    (4+3) + (3+2) + (2+1) + 1

    Please do the question and see if your answer matches up with my 2nd answer.
    I get 31 unique tastes.
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  3. #3
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    Well It should be  31

    31 = {5 \choose 1}+ {5\choose 2}+{5\choose 3}+{5\choose 4}+{5\choose 5}

    (Is "no taste" a taste too? ;p)
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  4. #4
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    Quote Originally Posted by Dinkydoe View Post
    (Is "no taste" a taste too? ;p)
    I thought about that, too.
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    Hello, Masterthief1324!

    I agree with skeeter and Dinkydoe . . .


    If the papillae (bumps on your tongue) can sense 5 basic tastes
    and any combination of those tastes produce a unique taste,
    how many possible unique tastes can you have?

    . . \begin{array}{ccccc}<br />
\text{1-at-a-time:} & _5C_1 &=& 5 \\<br />
\text{2-at-a-time:} & _5C_2 &=& 10 \\<br />
\text{3-at-a-time:} & _5C_3 &=& 10 \\<br />
\text{4-at-a-time:} & _5C_4 &=& 5 \\<br />
\text{5-at-a-time:} & _5C_5 &=& 1 \\ \hline<br />
& \text{Total:} && {\color{blue}31}<br />
\end{array}



    Call the five basic tastes: a,b,c,d,e
    You can list them and count them yourself . . .

    . . \begin{array}{ccc}\text{1-at-a-time:} & a,b,c,d,e \\ \\<br />
\text{2-at-a-time:} & ab,ac,ad,ae,bc\\<br />
& bd,be,cd, ce, de \\ \\<br /> <br />
\text{3-at-a-time:} & abc, abd, abe, acd, ace \\<br />
& ade, bcd, bce, bde, cde \\ \\<br /> <br />
\text{4-at-a-time:} & abcd, abce, abde, acde, bcde \\ \\<br />
\text{5-at-a-time:} & abcde<br />
\end{array}



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  6. #6
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    Quote Originally Posted by Soroban View Post
    Hello, Masterthief1324!

    I agree with skeeter and Dinkydoe . . .


    . . \begin{array}{ccccc} \text{1-at-a-time:} & _5C_1 &=& 5 \\
    \text{2-at-a-time:} & _5C_2 &=& 10 \\
    \text{3-at-a-time:} & _5C_3 &=& 10 \\
    \text{4-at-a-time:} & _5C_4 &=& 5 \\
    \text{5-at-a-time:} & _5C_5 &=& 1 \\ \hline
    & \text{Total:} && {\color{blue}31}
    \end{array}" alt="
    \text{1-at-a-time:} & _5C_1 &=& 5 \\
    \text{2-at-a-time:} & _5C_2 &=& 10 \\
    \text{3-at-a-time:} & _5C_3 &=& 10 \\
    \text{4-at-a-time:} & _5C_4 &=& 5 \\
    \text{5-at-a-time:} & _5C_5 &=& 1 \\ \hline
    & \text{Total:} && {\color{blue}31}
    \end{array}" />



    Call the five basic tastes: a,b,c,d,e
    You can list them and count them yourself . . .

    . . \begin{array}{ccc}\text{1-at-a-time:} & a,b,c,d,e \\ \\ \text{2-at-a-time:} & ab,ac,ad,ae,bc\\
    & bd,be,cd, ce, de \\ \\

    \text{3-at-a-time:} & abc, abd, abe, acd, ace \\
    & ade, bcd, bce, bde, cde \\ \\

    \text{4-at-a-time:} & abcd, abce, abde, acde, bcde \\ \\
    \text{5-at-a-time:} & abcde
    \end{array}" alt="
    \text{2-at-a-time:} & ab,ac,ad,ae,bc\\
    & bd,be,cd, ce, de \\ \\

    \text{3-at-a-time:} & abc, abd, abe, acd, ace \\
    & ade, bcd, bce, bde, cde \\ \\

    \text{4-at-a-time:} & abcd, abce, abde, acde, bcde \\ \\
    \text{5-at-a-time:} & abcde
    \end{array}" />


    I will familiarize myself with the combination and permutation notation later.
    If I was to write down all the possible combinations, what is the best way to keep track of the combinations so as to avoid overlapping combinations?
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  7. #7
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    Quote Originally Posted by Masterthief1324 View Post
    I will familiarize myself with the combination and permutation notation later.
    If I was to write down all the possible combinations, what is the best way to keep track of the combinations so as to avoid overlapping combinations?
    No "best" way ... just be organized.

    Soroban's listing is very organized.
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  8. #8
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    Here is another way to arrive at the same answer.

    Each of the 5 tastes is either present or not present-- 2 choices. So there are

    2^5

    possibilities. But one of those is the "no taste" combination. If we eliminate that one, there are

    2^5 -1 = 31

    possibilities.
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