In how many ways 11 identical marbles be placed in 3 distinct jars suc

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