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If a train departs from Kota to Delhi having 12 enroute station then find the number of ways in which the train can be stopped at four station which are not consecutive?

If a train departs from Kota to Delhi having 12 enroute station then find the number of ways in which the train can be stopped at four station which are not consecutive?

Grade:12th pass

1 Answers

ABHIJEET SHUKLA
30 Points
3 years ago
 
hmm sunil no issues..i ll try to explain..

consider a+b+c=10   where a,b,c >=0
this can be done in  (10+2) C 2  ie 12C2..here 2 is no of plus signs

consider a+b+c+d =12  so this can be done in 15C3

Now this was when a,c,b..terms were >=0
but if we know any other lower limit for any term then wat

a+b+c=10  a,b >=0 and c >=1

here now i ll give 1 to c and change limit of c

ie a+b+c=10
           1=9
--------------
now a+b+c=9 and a,b,c>=0  so no. of ways is (9+2)C2 ie 11C2
----------------------------this was jus to explain to find no of ways-----------
Now our question:

here stations are A   a  b  c  d  e  f  g  h  i  j  k  l         B

now we know train will halt at any 4 stations among a b c ...k l and these halting stations should not be consecutive.So wat should we do to achieve this...
A      a     b    c         e  f    g      h   i    j     k   l             B
       [ M ]     [ N ]       [  O ]       [    P ]        [Q    ]

see in above description..
M is no of stations before 1st halting station..
N is no of stations between 1st and 2nd halting stations
O is no of stations between 2nd and 3rd halting stations 
P is no of stations between 3nd and 4th halting stations 
Q is no of stations between 4th and 5th halting stations 

also note that to satisfy our condition..M can be 0..since it can halt at 1st station itslf..similarly Q can also be O..coz train can always halt at last stattion

but here N,O,P should be atleast 1 to make halting stations non consecutive..

also we know M+N+O+P+Q=12-4   (4 halting stations removed) 
             and M,Q >=0  and N,O,P>=1

so M+N+O+P+Q=8
        1   1  1     5
------------------------
M+N+O+P+Q=5  and M,N,O,P,Q >=0

so no of ways are  (5+4)C4  ..9C4.         here 4 is again no. of plus signs as i mentioned earlier.
INSTEAD OF KOTA AAN DELHI IT IS A NAD B

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