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)}= 12C_{2} 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".**

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among four kids so that:
a) each kid gets at least 1 apple
b) each kid gets an even number of apples but no kid gets 0
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1. How many ways are there to distribute 12 identical apples
among 5 baskets, with the last two baskets having 4 apples
combined?

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there is no constraint?
first urn has at least 1 red ball and 2 blue balls?
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