Anyone able to explain clearly on labelling? If possible, expand to factorial and combination/permutation as well? I am able to calculate factorial, combination and permutation but I reckon that I would better grasp an understanding of their logic rather than relying on calculation. Thanks!
labelling is, in how many number of ways you can assign some given labels to ALL members in a group, when you know, to exactly how many members you need to assign each label. For example, you have a portfolio of 10 stocks (there are 10 members in the group). You need to assign 3 labels to them. And these labels are ‘Buy’, ‘Sell’ and ‘Hold’. Also, you know that you need to assign ‘Buy’ lable to exactly 2 stocks, ‘Sell’ label to exactly 3 stocks and ‘Hold’ label to exactly 5 stocks. And the question is, in how many ways you can do this? As per the equation, the total number of ways are n! / (n1! x n2! x n3!), which is 10! / (2! x 3! x 5!) The logic is, 10! is the total number of ways these 10 stocks can be arranged without any restrictions. Then you divide this by number of given restrictions to get your answer. Actually, I think this logic is coming from ‘Combination’. Combination will be used here, as the order is not important. By Combination’s logic: 1. First choose 2 stocks to label as ‘Buy’ from a list of 10 stocks. That will be 10C2. 2. next, choose 3 stocks to label as ‘Sell’ from a list of remaining 8. That will be 8C3. 3. last, choose 5 stocks to label as ‘Hold’ from a list of remaining 5. That will be 5C5. So, your result is 10C2 x 8C3 x 5C5, which if you expand would be same as 10! / (2! x 3! x 5!). Does it help?
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Thanks rus1bus, it does helps a little. Why do we have to divide by (n1! x n2! x n3!) in order to label them into 3 different groups? I am getting brood down by the maths.
the n1! is the number of ways they can internally be rearranged… you have two things A and B -> they can be arranged as AB or BA but from your perspective for labelling - both are the same whether AB or BA. So you need to divide by 2 otherwise you would be getting a higher number of labels… not sure if I am explaining it correctly… but you get the idea, I hope.