a+b+c = 11

Total positive integer solutions = (n-1)C(r-1)

Here, n = 11, r = 3

So total Integer solutions = (11-1)C(3-1) = 10C2 = 45

Answer: Option C
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I have been doing such problems by applying 2 equations.
1) when we can distribute 0- n things to r boxes , no of ways = (n+r-1)C(r-1)
2) when we are not allowed to keep any box empty =(n-1)C(r-1)
Applying the second formula, 10C2 =45.
C is the answer.

It would be appreciated if someone can elaborate those two formulas.

Let the number of balls in the jars be a, b, c, where none of the numbers are zero

=> a + b + c = 11

Using formula: Since we need positive integer solutions, we have: (11-1)C(3-1) = 10C2 = 45 ways

Alternative formula: We assign 1 marble to each of a, b and c; now these are already positive. Thus, we have to distribute 8 more marbles among those 3; but now, the distribution allows for 0 marbles assigned => Number of ways = (8+3-1)C(3-1) = 10C2 = 45 ways
[Thus, it is sufficient to simply know the second formula]



Explanation - How do we get these formulae?

Let the 11 identical marbles be denoted as M M M M M M M M M M M
We need to put 2 partitions so that we divide the marbles in 3 groups, say for example: M M M | M M M M | M M M M (where | represents the partition)

Let us represent the above as: M M M P M M M M P M M M M

This is a 13 letter word with 11 Ms and 2 Ps and if we arrange it, we will get all distributions based on the position of the Ps

Ex:
M M M P M M M M P M M M M => 3, 4, 4 distribution
M M M M M M M M M M P M P => 10, 1, 0 distribution; and so on

Note: assigning 0 to a group is allowed

Thus, total arrangements = 13!/(11!2!) = 13C2 = (11+3-1)C(3-1)

Thus, non-negative integer solutions for a+b+c+d+... (r groups) = N is: (N+r-1)C(r-1) where N represents identical objects and r represents distinct groups



Let us now analyse "positive integer solutions":

Continuing in the same manner as above:

Let the 11 identical marbles be denoted as M M M M M M M M M M M
We need to put 2 partitions so that we divide the marbles in 3 groups

To ensure that no group ends up getting zero marbles, we assign 1 marble to each group first, leaving us with 8 marbles. We need to divide the marbles in 3 groups, say for example: M M M | M M M M | M (where | represents the partition)

Let us represent the above as: M M M P M M M M P M

This is a 10 letter word with 8 Ms and 2 Ps and if we arrange it, we will get all distributions based on the position of the Ps

Thus, total arrangements = 10!/(8!2!) = 10C2 = (11-1)C(3-1)

Thus, positive integer solutions for a+b+c+d+... (r groups) = N is: (N-1)C(r-1) where N represents identical objects and r represents distinct groups
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I've always been using only (n+r-1)C(r-1) but when no box can be left empty, I put one marble in each box and apply the same formula for what is left.

So in this case I put one marble in each jar so I'm left with 8 and so n=8, r=3, (n+r-1)C(r-1) = 10C2=45

Posted from my mobile device
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minustark

Please watch the following video in which I have explained where does this property come from

I hope it help!

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