106of course there is ..
NON NEGATIVE ≠POSITIVE
Consider these two questions-
1) find the number of non negative integral solutions of x+y+z+w = 20
2) find the number of positive integral solutions of x+y+z+w = 20
answer to 1) is 23C3 and 2) is 19C3
Is there any difference between the two questions? well I dont think so.
What is the correct answer then?
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14 Answers
301) Since solutions are non-negative, 0 is also a possible solution.
Maximum no. of zeroes that can be used = 3
So we arrange 20 1s and 3 zeroes.
1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_0_0_0_
Now to get solutions we just have to put 3 plus (+) signs in the given 23 slots.
Hence no of solutions 23C3
2) For positive solutions, x,y,z,w > 0
Hence we arrange 20 1s.
1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1
We have to put 3 plus(+) signs to get the solutions.
Hence, no of solutions = 19C3
Source : Nishant Sir [3] [6]
30@Khyati : Since we have four numbers to add, putting three plus signs in those slots would act like three dividers. Hence we get four distinct numbers and the arrangements ofl these plus signs gives us all the possible solutions!
Btw, 1+1 uses 1 one plus sign. 1+2+3 uses 2 and 1+2+3+4 uses 3. You could look at it either way! [1] [3]
4920 1s and 3 zeroes gives 23
but the no of places in whch we cn put + is 22
so why is the answer not 22C3???
49however...we can do it in this way...
1)
we have to put 3 plus es between 20 1s
so what we do take a boz and put 20 1s and 3 pluses and start pick out any 3... if these 3 are the pluses that will give us the soln...
so the soln is 23C3
2)
x+y+z+w=20
let x=1+a
y=1+b
z=1+c
w=1+d
such a,b,c,d≥0
whch makes x,y,z,w≥1
so...we cn write..
1+a+1+b+1+c+1+d=20
or, a+b+c+d=16
now number of solution in this case is equal to number of soln to the original eqn...
so by previous logic, no of solutions is 19C3
11@ Subhomoy,
the method given by Ashish is much easier and it will take less time also and prevent
confusion too.(Yeah I agree it should be understood well to where take zeros and where
not and slots sng (+) signs [3])
30Subhomoy,
I like the box idea! [3] But, I dont understand how we get different solutions because we might as well just pick 3 ones out of the 23 things. [7] Or maybe 2 1s and a plus sign.
1Sorry to say, But the pinked explanation is not correct.
Although it may provide the correct answer.
Explanation for the case of positive values is correct but for non negative ones ...not so.
Because:
@Ashish ...
1_1_1_1_1_0_0_
for x+y+z=5
putting pluses in gaps between them is like assigning values to x y and z. But as you can see x will never get a value zero assigned to it.
So the method is incorrect.
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62@Mkagenius...
You are right..
the logic asish has given in the first part is not correct...
what he should have done is put 3 +'s in between 20 1's
so it is arranging 23 object (3 of one kind +) and 20 of another (1)
