docabuzar wrote:
Many Thnx.
2 similar Qs.
1. In how many ways can 5 Idnetical fruits be distributed in 4 identical baskets?
Fruits are identical but baskets are also identical so we cannot apply 8C3 from n+r-1 C r-1?
Can we say that to remove the duplications of baskets we should divide 8C3 by 4! ?
2. In how many ways can 5 different fruits be distributed in 4 identical baskets?
Is this like making 4 groups from 5 different fruits so = 5!
I think you are overthinking the concept by introducing the "identical basket" situation. For the first case if, for example, there is no difference between the following scenarios: {5-0-0-0} and {0-5-0-0}, then you can manually write down all possible cases: {5-0-0-0}, {4-1-0-0}, {3-1-1-0}, {3-2-0-0}, {2-2-1-0}, {2-1-1-1}. So the answer for the first question as you stated would be 6.
Don't overcomplicate it: the GMAT combination/probability questions are fairly straightforward and no need to waste time on the problems you will never see on the test.
If the questions were:
1. In how many ways can 5 identical fruits be distributed in 4 different baskets?
Consider five stars and three bars: *****|||, where stars represent fruits and three bars (one less than the baskets) will help us to divide them among the baskets. How many permutations (arrangements) of these 5+3=8 symbols are there? Permutation of 8 symbols out of which 5 * and 3 | are identical is 8!/(5!3!). Now, each arrangement will mean different scenario for fruit distribution. For example: **|*|*|* will mean that the first basket gets 2 fruits, the second, third and fourth 1 fruit. Or: ***|*|| will mean that the first basket gets 3 fruits, the second 1, and third and fourth none, and so on.Answer: 8!/(5!3!).
2. In how many ways can 5 different fruits be distributed in different 4 baskets?This one is easier each of the 5 different fruits has 4 choices, so total # of distribution is 4*4*4*4*4=4^5.
Answer: 4^5.
Check similar questions:
how-many-positive-integers-less-than-10-000-are-there-in-85291.htmlcombinations-tough-108739.htmlsolve-these-gmat-question-98701.htmlvoucher-98225.htmlTheory on permutation and combinations:
permutation-86687.htmlDirect formula if needed:
The total number of ways of dividing n identical items among r persons, each one of whom, can receive 0,1,2 or more items is \(n+r-1C_{r-1}\).
The total number of ways of dividing n identical items among r persons, each one of whom receives at least one item is \(n-1C_{r-1}\).Hope it helps.
Why cant we use the same logic as used in the ring problem? How does using identical and different change the solution? I mean these are not word problems where the combination has to be unique?