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Grade: 12th pass
        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?
2 years ago

Answers : (1)

ABHIJEET SHUKLA
30 Points
							
 
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
2 years ago
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