Discrete mathematics - Combinatorics

Question

An integer is called snakelike if its decimal representation  $a_1a_2a_3...a_n$ satisfies $a_i < a_{i+1} $  if $i$ is odd and $a_i > a_{i+1}$ if $i$ is even. How many snakelike integers between $1000$ and $9999$ have four distinct digits?

Comments

Nice question I should say..:)

Answer 1

Here we need to decompose the problem into 2 subproblems :

a) Numbers which do not include zero in any of its digits..

Let we have digits w , x , y and z such that  w < x < y < z..This can be done in ⁹C₄ ways

Now according to the definition of snake like integers , the possible permutation of these digits are :

i) w y x z       ii)  w z x y    iii)   x y w z       iv) x z w y    v) y z w x

Hence number of valid permutations for the given selection of 4 digits  =  5

Thus total number of ways under this case    =   ⁹C₄ * 5

b) Numbers which include 0 :

Here we need to choose only three digits out of 9 which can be done in ⁹C₃ ways..

Let the digits are x , y and z such that 0 < x < y < z

Now valid permutations for each selection :

i) x y 0 z     ii)  x z 0 y     iii) y z 0 x

Hence total number of ways in this case                              =       ⁹C₃ * 3 ways

Thus total snake like numbers  having 4 digits                    =       ⁹C₄ * 5   + ⁹C₃ * 3

                                                                                            =       882

Comments

I get confused while writing the possible permutations. is there any methodical way ?

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There is no such specific method I think..:)..U have to check according to the definition.

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Thank you so much for helping @Habibkhan