21mak do u mean to say LRFB BBBBB
& RLFB BBBBB
WILL BE COUNTED IN THE SAME CASE????
COZ DA QUE ASKS IN HOW NMANY WAYS CAN HE TAKE 9 STEPS SO I THINK THEY SHUD B DIFF YAAAARRRR!!!!!!!!!
SO DA ANS SHUD BE (9P4 * 4^5)
pls ans dis ques
A man has to take 9 steps. he can move in 4 directions left, right, forward, backward.
1. in how many ways he can move 9 steps??
2. in how many ways he can move 9 steps such that he has to take steps in all directions (at least one step in every direction).
3. in how many ways he can move 9 steps such that after finishing his 9 steps he is one step away from starting position....
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31 Answers
21
1for 1st its 49.
for 2nd its 4*3*2*1*9c4*45
for third i haven't calculated but nishant bhaiya is right.
plz post the ans i will calculate the third and check my ans.
13but why....???????
i dont agree unless someone points out my mistake... anywayzzz... sorry, if i'm wrong... [2]
62for part 3
take that no of lefts = L, Rights =R, Top=T, bottoms=B
L+R+T+B=9
Case 1, L=R, T=B+1
Case 2, L=R, T=B-1
Case 3, L=R+1, T=B
Case 4, L=R-1, T=B
Now take this ahead... Mak I am not sure if what you have done for this part is right either....
13i couldn't find any mistake in second one... [2]
can u plzzz point it out...
13The order of those steps taken doesn't matter... in d question, it is juzz said dat atleast one step is to be taken in each direction... so order doesn't matter here...
[1]
21Hey mak
In part 2 of the question
dont u thik da ans shud be 9P4
bcoz those directions selected can be in any arrangement!!!!
13from the above explanation, u may raise a doubt .... "If dat is d case, then multiplying 220 with 4! (since each case gives 4! new cases) must give d desired answer, but it doesn't. y...? "
this is not true because there are many more possibilities... not just 4! , but many moreeeee...!!!!!
[infact i got this doubt while posting d above reply, dunno whether u'll get dis doubt or not... [6] [4] ]
13@shreya...
in ur method, the error is... :
u've considered all d 9 steps as identical... where as, they are all different...
more precisely... in ur equation x1 + x2 + x3 + x4 = 9... if v solve by ur method, v assume dat (0,2,3,4) and (2,3,4,0) are same... where as, here they are different solutions...
practically speaking, taking 2 steps in north direction and one step in west direction is not equal to taking one step north, one west and then one north... though both of them will land u up at d same point...
hope its clear now... [1]
13Explanations for my answers...
1) As already said by nishant bhaiya, each individual step can be taken in 4 ways irrespective of d previous n further steps...
hence the total no. of ways = 4.4.4...(9 times) = 49 ways...
2) Selecting four of d total nine steps in 9C4 ways... and it can be assumed that each of the four steps are taken in four different directions... now the remaining 5 steps can be taken in any direction in 45 ways...
hence total no. of such cases possible = 9C4.45 ways...
3) This one is interesting...
First step can be taken in any direction in 4 ways... now, for each step taken later on, one counter step should be taken... only then it is possible for him to be one step away from starting point... (think over this, it's d easiest way to solve dis problem...!!!) hence if v decide any of the four steps, the other four steps (since total steps remaining after first step are 8) will be counter steps to these four... [note : counter step means a step taken in d opposite direction to d step already taken... its not necessary that same step should be reversed, but it must be in d opposite direction, or rather... a counter step means an anti-parallel step]
hence, no. of ways this can be done = 4.44 = 45 ways...
[1]
correct me if i've gone wrong anywhere in my interpretation...
62shreya.. for part 1, the thing that you have done is not exactly what needs to be done..
I will tell u my proof.. then you can think if u understand why your proof has a mistake....
at each step, the man can take one of the 4 directions.. which is independent of all his previous motions... So for each of the 9 steps he has 4 options..
1i hve done da first part... n i m nt gettin 49..
my ans is 220 ways...
solved it by integral equation method
it will be x1 + x2 + x3 + x4 = 9
constraints will be 0<=x1<=9, 0<=x2<=9, 0<=x3<=9, 0<=x4<=9..
and on solving dis u get coeffiecient of x9 in (1-x)-4
so ans is 12C9. ie 220 ways..
i think for 2nd part also we can solve by dis method n in 3rd part we would hve to make cases...
do correct me if i m wrong....
13For d first one... ans is 49
For d second one... ans is 9C4.45
For d third one... ans is 4.(44)
[1]
correct me if i'm wrong...
62guys.. give the final answers.. i will post if right or wrong...
THis is a good question :)
62yup these two cases will suffice :)
one when n=m for left and righ
2nd when n=m for top and below..
1can we do da 3rd part lik dis..
we want steps in forward direction= dat in backward direction
n steps in left = steps in right
n then we make cases
62and yeah... i din post the full solution so that u can try on ur own.. :)
622nd part u have to solve by taking cases... (i couldnt think of an easier method)
3rd part is easier.... thikin in terms of n left , m right .. so what will be the relation between m and n?
3oh edited shreya the first step can be done in 4 ways and so the 9 steps can be done independently so total ways 4*4*4*4*4*4*4*4*4 ways
62!)
he has 4 option at all times ... independent of this previous ones..
so 49