Question

How many ways can you distribute 15 identical marble into 6 distinguishable urns so that the sum total of marbles in the first three urns equals 5?

Answer 1

Possibilities for sum of 5 are: (1,1,3), (3,1,1), (1,3,1),(1,2,2), (2,2,1),(2,1,2), i.e., 6 ways. ​​​​​​

After filling 3 urns, we are left with 6-3 =3 urns that are to filled with remaining 15-5 =10 identical marbles.

Formula: n identical items are distributed among r boxes(urns) is: (n+r-1)C_(r-1) ways.

r =3 urns and n =10 identical marbles.

Thus, (10+3–1)C_(3-1)= 12C₂ ways =(12*11)/2= 66 ways. ​​​​​​

Therefore, total number of ways =6*66 =396 ways.

So, you can distribute 15 identical marbles into 6 distinguishable urns so that the sum total of marbles in the first three urns equals 5 is "396 ways".