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Math Help - in how many ways?

  1. #1
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    i live on the first floor of my building. everyday we come back after a walk me and my dog need to climb 7 stairs. forget about me, how many ways can my dog climb them if he climbs 1-2 stairs in one step.



    eg. he may either climb the 7 stairs by climbing 2, then 2, then 2, then 1, or 1, then 2, then 2, then 2, or he may climb then as 2,2,1,1,1........in total, how may possible ways? thats all i mean.
    Last edited by mr fantastic; May 7th 2009 at 04:50 AM. Reason: Merged posts, minor editing
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  2. #2
    Newbie scouser's Avatar
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    There are two outcomes to each event: the solution is 2^7. there. figure that out, i dont have my calculator on me.
    Last edited by mr fantastic; May 7th 2009 at 04:59 AM. Reason: The thread was unacceptable - like something from the Chatroom.
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  3. #3
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    mr fantastic's Avatar
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    Quote Originally Posted by scouser View Post
    There are two outcomes to each event: the solution is 2^7. there. figure that out, i dont have my calculator on me.
    This is not correct (and you need a calculator? 2^7 = 2^3 x 2^3 x 2 = 8 x 8 x 2 = 64 x 2 = 128)

    The possible combinations are:

    (2, 2, 2, 1)

    (2, 2, 1, 1, 1)

    (2, 1, 1, 1, 1, 1)

    (1, 1, 1, 1, 1, 1, 1)

    You have to calculate the number of distinct arrangements of each combination and add them together:

    \frac{4!}{3!} + \frac{5!}{2! 3!} + \frac{6!}{5!} + 1 = ....
    Last edited by mr fantastic; May 7th 2009 at 05:08 AM.
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  4. #4
    Senior Member pankaj's Avatar
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    Let f(n) denote the number of ways to climb n stairs.

    Therefore, f(n-2)+f(n-1)=f(n)

    Now f(1)=1
    f(2)=2
     <br />
f(3)=f(1)+f(2)=3<br />
    f(4)=f(3)+f(2)=5
    ................... and so on
    f(7)=8+13=21

    Therefore number of ways tho dog can climb 7 stairs in 21 ways.
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  5. #5
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    Quote Originally Posted by pankaj View Post
    Let f(n) denote the number of ways to climb n stairs.

    Therefore, f(n-2)+f(n-1)=f(n)

    Now f(1)=1
    f(2)=2
     <br />
f(3)=f(1)+f(2)=3<br />
    f(4)=f(3)+f(2)=5
    ................... and so on
    f(7)=8+13=21

    Therefore number of ways tho dog can climb 7 stairs in 21 ways.
    hey! thanks. but yeh neeche likhi wali equation kaise ayi?

    f(n-2)+f(n-1)=f(n)
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  6. #6
    Senior Member pankaj's Avatar
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    There are two ways in which one can reach the n-th stair(to be done in f(n) ways).

    One way is that one reaches the (n-2)th stair which can be done in f(n-2) ways and then reach the n-th stair by climbing 2 steps in one go.

    Second way is that one reaches the (n-1)th stair which can be done in f(n-1) ways and then reach the n-th stair by climbing the remaining 1 stair.

    Therefore, f(n)=f(n-1)+f(n-2)
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